Kuliah Diff Eq WS

172
DIFFERENTIAL EQUATIONS (DE)

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Transcript of Kuliah Diff Eq WS

Page 1: Kuliah Diff Eq WS

DIFFERENTIAL EQUATIONS(DE)

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DEFINITION: differential equation

An equation containing the derivative of one or more dependent variables, with respect to one or more independent variables is said to be a differential equation (DE).

Definitions and Terminology

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Differential Equations

A differential equation is an algebraic equation that contains some derivatives:

037352

2

ydx

dy

dx

ydty

dt

dy

• Recall that a derivative indicates a change in a dependent variable with respect to an independent variable.

• In these two examples, y is the dependent variable and t and x are the independent variables, respectively.

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Why study differential equations?

• Many descriptions of natural phenomena are relationships (equations) involving the rates at which things happen (derivatives).

• Equations containing derivatives are called differential equations.

• Ergo, to investigate problems in many fields of science and technology, we need to know something about differential equations.

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Why study differential equations?

• Some examples of fields using differentialequations in their analysis include:

— Solid mechanics & motion— heat transfer & energy balances— vibrational dynamics & seismology— aerodynamics & fluid dynamics— electronics & circuit design— population dynamics & biological systems— climatology and environmental analysis— options trading & economics

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Examples of Fields Using Differential Equations in Their Analysis

CoolBath

HotBath

kydt

dyckx

dt

dxc

dt

xdm

2

2

02

2

kxdt

dxc

dt

xdm

ikkxdt

dxc

dt

xdm f

2

2

ghACdt

dhA od 2

70 Hkdt

dH

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Recall Calculus

Definition of a Derivative

If , the derivative of or

With respect to is defined as

The derivative is also denoted by or

Definitions and Terminology

)(xfy y )(xfx

h

xfhxf

dx

dyh

)()(lim

0

dx

dfy ,' )(' xf

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Recall the Exponential function

dependent variable: y

independent variable: x

Definitions and Terminology

xexfy 2)(

yedx

xde

dx

ed

dx

dy xxx

22)2()( 22

2

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Differential Equation: Equations that involve dependent variables and their derivatives with respect to the independentvariables.

Differential Equations are classified bytype, order and linearity.

Definitions and Terminology

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Differential Equations are classified bytype, order and linearity.

TYPE

There are two main types of differential equation: “ordinary” and “partial”.

Definitions and Terminology

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Ordinary differential equation (ODE) Differential equations that involve only ONE independent variable are called ordinary differential equations.

Examples:

, , and

only ordinary (or total ) derivatives

Definitions and Terminology

xeydx

dy5 06

2

2

ydx

dy

dx

yd yxdt

dy

dt

dx 2

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Definition:

A differential equation is an equation containing an unknown function and its derivatives.

32 xdx

dy

032

2

aydx

dy

dx

yd

364

3

3

y

dx

dy

dx

yd

Examples:.

y is dependent variable and x is independent variable, and these are ordinary differential equations

1.

2.

3.

ordinary differential equations

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Partial differential equation (PDE)Differential equations that involve two or more independent variables are called partial differential equations.Examples:

and

only partial derivatives

Definitions and Terminology

t

u

t

u

x

u

22

2

2

2

x

v

y

u

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Partial Differential Equation

Examples: 02

2

2

2

y

u

x

u

04

4

4

4

t

u

x

u

t

u

t

u

x

u

2

2

2

2

u is dependent variable and x and y are independent variables, and is partial differential equation.

u is dependent variable and x and t are independent variables

1.

2.

3.

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Derivatives

Derivatives

dx

dy

Partial Derivatives

u is a function of more than one

independent variable

Ordinary Derivatives

y is a function of one independent variable

y

u

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Differential Equations

• Heat transfer

• Mass transfer

• Conservation of momentum, thermal energy or mass

dz

TCd

A

q pz)(

dz

dCDJ A

ABAZ

Rzt

2

2

ODE

PDE

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ORDER

The order of a differential equation is the order of the highest derivative found in the DE.

second order first order

Definitions and Terminology

xeydx

dy

dx

yd

45

3

2

2

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Definitions and Terminology

xeyxy 2'

3'' xy

0),,( ' yyxFfirst order

second order 0),,,( ''' yyyxF

Written in differential form: 0),(),( dyyxNdxyxM

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LINEAR or NONLINEAR

An n-th order differential equation is said to be linear if the function

is linear in the variables

there are no multiplications among dependent variables and their derivatives. All coefficients are functions of independent variables.

A nonlinear ODE is one that is not linear, i.e. does not have the above form.

Definitions and Terminology

)1(' ,..., nyyy

)()()(...)()( 011

1

1 xgyxadx

dyxa

dx

ydxa

dx

ydxa

n

n

nn

n

n

0),......,,( )(' nyyyxF

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LINEAR or NONLINEAR

or

linear first-order ordinary differential equation

linear second-order ordinary differential equation

linear third-order ordinary differential equation

Definitions and Terminology

0)(4 xydx

dyx

02 ''' yyy

04)( xdydxxy

xeydx

dyx

dx

yd 53

3

3

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LINEAR or NONLINEAR

coefficient depends on y

nonlinear first-order ordinary differential equation

nonlinear function of y

nonlinear second-order ordinary differential equation

power not 1

nonlinear fourth-order ordinary differential equation

Definitions and Terminology

0)sin(2

2

ydx

yd

xeyyy 2)1( '

024

4

ydx

yd

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xeydx

dy

dx

yd x cos44)( 23

3

ordinary differential equation :only one independent variable involved: x

)(2

2

2

2

2

2

z

T

y

T

x

Tk

TC p

partial differential equation: more than one independent variable involved: x, y, z,

Differential Equation

An equation relating a dependent variable to one or more independent variables by means of its differential coefficients with respect to the independent variables is called a “differential equation”.

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xeydx

dy

dx

yd x cos44)( 23

3

3rd order O.D.E.

1st degree O.D.E.

Order and Degree

The order of a differential equation is equal to the order of the highest differential coefficient that it contains. The degree of a differential equation is the highest power of the highest order differential coefficient that the equation contains after it has beenrationalized.

xeydx

dy x cos44)( 2 1st order O.D.E.

2nd degree O.D.E.

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xeydx

dy

dx

yd x cos44)( 23

3

product between two derivatives ---- non-linear

xydx

dycos4 2

product between the dependent variable themselves ---- non-linear

Linear or Non-Linear

Differential equations are said to be non-linear if any products exist between the dependent variable and its derivatives, or between thederivatives themselves.

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Terminologies:1) Ordinary vs. partial differential equationsAn ordinary differential equation (ODE) involves only one independent variable and contains only total differentials.

A partial differential equation (PDE) involves more than one independent variable and contains partial differentials.

2) Linear vs. nonlinear differential equationsA linear differential equation contains only terms that are linear in the dependent variable or its derivatives.

A nonlinear differential equation contains nonlinear function of the dependent variable.

First order differential equations

Differential equations

xxyxdx

xdy4)(

)( 2

0),(),(

5),(

yxy

yxxy

x

yx

xxyxdx

xdy4)(

)( 2

ydx

xdyx

dx

xydxy

dx

xdyx

dx

xydxyx

dx

xydsin

)()( ,0)(

)()( ,0)(

)( 22

22

2

222

2

2

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3) Order of a differential equationThe order of a differential equation is determined by the highest derivative in the equation.

4) Homogeneous vs. inhomogeneous differential equationsA linear differential equation is homogeneous if every term contains the dependent variable or its derivatives.

• A homogeneous differential equation can be written as where L is a linear differential operator.

• A homogeneous differential equation always has a trivial solution y(x) = 0.• Superposition principle: If y1(x) and y2(x) are solutions to a linear homogeneous

differential equation, then ay1(x) +by2(x) is a solution to the same equation.

order second ,0)(4)()(

orderfirst ,4)()(

22

2

2

xxydx

xdyx

dx

xyd

xxyxdx

xdy

0)(4)()( 2

2

2

xxydx

xdyx

dx

xyd

,0)( xyL

xdx

dx

dx

d4 e.g., 2

2

2

L

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An inhomogeneous differential equation has at least one term that contains no dependent variable.

• Inhomogeneous equations are often called “driven” equations. They describe the response of the system to an external force.

• The general solution to a linear inhomogeneous differential equation can be written as the

sum of two parts:

Here yh(x) is the general solution of the corresponding homogeneous equation, and yp(x) is any particular solution of the inhomogeneous equation.

xxyxdx

xdy4)(

)( 2

tAxxx sin20

)()()( xyxyxy ph

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Solutions of ODEs

DEFINITION: solution of an ODEAny function , defined on an interval I and possessing at

least n derivatives that are continuous

on I, which when substituted into an n-th order ODE reduces the equation to an identity, is said to be a solution of the equation on the interval.

(Zill, Definition 1.1, page 8).

Definitions and Terminology

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Namely, a solution of an n-th order ODE is a function which possesses at least n

derivatives and for which

for all x in I

We say that satisfies the differential equation on I.

Definitions and Terminology

0))(),(),(,( )(' xxxxF n

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Verification of a solution by substitution

Example:

left hand side:

right-hand side: 0

The DE possesses the constant y=0 trivial solution

Definitions and Terminology

xxxx exeyexey 2, '''

xxeyyyy ;02 '''

0)(2)2(2 ''' xxxxx xeexeexeyyy

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ODE

• Definition

• Example

• A 3rd order differential equation for r = r(t)• Solution

independent

dependent

0)(),...,('),(, )( ttttf n (4.4)

4'''''' 2 tet (4.5)

kQdt

dQ (4.6)

tcetQ kt ,)( (4.7)

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Linear Equation (1)

1. Rewrite 4.9

2. Determine

where m(t) is called an integrating factor

)()()()(...)()( 0)0(

1'

2)1(

1)( tgtatatatata n

nn

n (4.8)

)()()(')( 012 tgtatata (4.9)

0)(;)(

)()(

)(

)(' 2

2

0

2

1

tata

tatg

ta

ta for all t (4.10)

dt

ta

tat

)(

)(exp)(

2

1 (4.11)

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Linear Equation (2)

3. Multiply both sides of equation 4.10 by m(t)

Observe that the left-hand side of eqn 4.12 can be written as

or

)(tdt

d

)()(

)()()(

)(

)('

2

0

2

1 tta

tatgt

ta

ta

(4.12)

)()(

)(exp

)(

)(

)(

)(exp'

2

1

2

1

2

1 tdt

ddt

ta

ta

ta

tadt

ta

ta

(4.13)

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Linear Equation (3)

Equation (4.12) can be rephrase as:

4. Integrate both sides of Equation (4.14) with respect to the independent variable:

dt

ta

ta

ta

tatgdt

ta

ta

dt

d

)(

)(exp

)(

)()(

)(

)(exp

2

1

2

0

2

1

cdtdtta

ta

ta

tatgdt

ta

ta

)(

)(exp

)(

)()(

)(

)(exp

2

1

2

0

2

1

(4.14)

dt

ta

tacdtdt

ta

ta

ta

tatgdt

ta

tat

)(

)(exp

)(

)(exp

)(

)()(

)(

)(exp)(

2

1

2

1

2

0

2

1 (4.15)

where c is the constant of integration

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Higher ODE Reduces to 1st Order2

2( ) ( )

Define , we have

( ) ( )

d y dyq x r x

dx dxdy

zdx

dyz

dxdz

r x q x zdx

2

1 2 3

12

23

231 3 2 1

'''( ) ( ) ''( ) 2( '( )) ( ) 0

Define , ', '', we have

2( )

y x y x y x y x y x

y y y y y y

dyy

dxdy

ydxdy

y y y ydx

In general, it is sufficient to solve first-order ordinary differential equations of the form

1( , , , ), 1,2, ,ii N

dyf x y y i N

dx

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• Nonlinear equations can be reduced to linear ones by a substitution. Example:

y’ + p(x)y = q(x)yn

and if n ¹ 0,1 then

n(x) = y1-n(x)

reduces the above equation to a linear equation.

(4.16)

(4.17)

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First Order Differential Equations

No general method of solutions of 1st O.D.E.s because of theirdifferent degrees of complexity.

Possible to classify them as:-exact equations

-equations in which the variables can be separated

-homogenous equations

-equations solvable by an integrating factor

First order linear differential equations occasionally arise in chemical engineering problems in the field of heat transfer,momentum transfer and mass transfer.

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First Order Ordinary Differential equation

38

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(1) The order of ODE: the order of the highest derivative

e.g.,

First-order ordinary differential equation

order) (second order),(first 2

2

dx

yd

dx

dy

(2) The degree of ODE: After the equation has been rationalized, the power of

the highest-order derivative.

e.g.,

ODE degree second andorder third the

)( and )()(0)( 23

3

3

332/322/3

3

3

dx

yd

dx

yd

dx

dy

dx

dyyx

dx

dyx

dx

yd

(3) The general solution of ODE contains constants of integration, that may

be determined by the boundary condition.

(4) Particular solution: The general solution contains the constants which are

found by the boundary condition.

(5) Singular solution: Solutions contain no arbitrary constants and cannot be

found from the general solution.

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with n parameters satisfies an nth-order ODE in general. The boundary conditions on the solutions determine the parameters.

xaxay cossin 21

First-order ordinary differential equation

General form of solution

),.....,,,,( 321 naaaaxfy

Ex: Consider the group of functions

equationorder -second 0

cossin

sincos

2

2

212

2

21

ydx

yd

xaxadx

yd

xaxadx

dy

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First-order ordinary differential equation

Exact equations

dy

ydFdxyxA

yyxB

y

yxU

yFyFdxyxAyxU

cyxU

yxdUx

yxB

y

yxA

y

yxU

xx

yxU

y

y

yxUyxB

x

yxUyxA

dyy

yxUdx

x

yxUyxdUdyyxBdxyxA

yxU

)(]),([),(

),(

by determined be can )( and )(),(),(

ODE of solution the is ),( so,

0),(),(),(

)),(

()),(

(

),(),( and

),(),(

),(),(),(),(),(

satisfies which ),( function aFor

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First-order ordinary differential equation

solution the is 2

3

2

3

)(0

)(2

3)()3(),(

exact is equation the1 ,1

),( ,3),(0)3(

03 :Ex

2

12

2

2

1

2

1

cxyx

ccyxx

cyFdy

dFx

dy

dFx

cyFyxx

cyFdxyxyxU

x

B

y

A

xyxByxyxAxdydxyx

yxdx

dyx

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Exact differential equations

0),(),( dyyxQdxyxP

A first-order ordinary differential equation can be generally written as

The differential is exact if we can find j (x, y) so that

).,( ),,(or ,0),(),( yxQy

yxPx

dyyxQdxyxPdyy

dxx

d

The solution of the differential equation is thus simply

The condition for the differential to be exact is

.)(),()(),(

)(),()(),(),(

yx

yx

CxgdttxQyfdtytP

xgdyyxQyfdxyxPyx

). ng(consideri . 2

yxx

Q

y

P

.2)(4)(23

exact. is aldifferenti the,4 Since

4

23

:1 Example

2322

22

Cxyxxgdyxyyfdxyx

yx

Q

y

P

xy

yx

dx

dy

yx

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0),(),( dyyxNdxyxM

dFdyy

Fdx

x

F

General solution: F (x,y) = C

For example

Exact Equations

Exact?

0 ydx

dyx Cxy

x

N

y

M

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Exact Equations Example

0)2(cossin3 dx

dyyxxyx

dyy

Fdx

x

FdF

xyxx

Fsin3

yxy

F2cos

)(cos4

1 4 yfxyxF

)('cos yfxy

F

')( 2 Cyyf Cyxyx 24 4cos4

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First-order differential equations

A first-order ordinary differential equation can be generally written as

)(

)(

yQ

xP

dx

dy

Separable variables

If the equation has the form , then

.0),(0)()(

0)()(

yxfdyyQdxxP

dyyQdxxPyx

),(

),(

yxQ

yxP

dx

dy

cxyydyxdx

y

x

dx

dy

yx

sin20cos

cos

:1 Example

2

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First-order ordinary differential equation

First-degree first-order equation

0),(),(or ),( dyyxBdxyxAyxFdx

dy

Separable-variable equation

dxxfyg

dyygxf

dx

dy)(

)()()(

1)2

exp(

)2

exp()2

exp(1

2)1ln(

1

)1( :Ex

2

22

2

xAy

xAC

xy

Cx

yxdxy

dy

yxxyxdx

dy

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For example

xdx

dyy sin

Separable-variables Equations

In the most simple first order differential equations, the independent variable and its differential can be separated from the dependent variable and its differential by the equality sign, using nothing morethan the normal processes of elementary algebra.

xdxydy sin Cxy cos2

1 2

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Example

1

93'

yx

yxySolve mYynXx �,Let�

)1(

)93(3

1)()(

9)()(3'

mnYX

mnYX

mYnX

mYnXY

01

093

mn

mn

3

2

m

nXY

XY

YX

YXY

/1

/33'

XYv /Let �

Xv'vY' v

vXv'v

1

3

X

dXdv

vv

v

223

1C'Xvv lnln)32ln(

2

1 2

C)3()3)(2(2)2(3 22 yyxx2

3

x

yv

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First-order ordinary differential equation

Homogeneous equations

x

dx

vvF

dv

vvFdx

dvxvF

dx

dvxv

dx

dy

vxy

x,yfyxf

yxf

yxBxyyxA

yxByxA

x

yF

yxB

yxA

dx

dy

n

)(

)()(

onsubstituti the Making

)(),( obeys

it , anyfor If n. degree shomogeneou of function a is ),( (2)

degree. third the with and e.g., degree,

same the of functions shomogeneou ),( and ),( Where(1)

)(),(

),(

3322

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x

yf

dx

dy

is termed a homogeneous differential equation of the first order.

Homogeneous Equations

Homogeneous/nearly homogeneous?

A differential equation of the type,

Such an equation can be solved by making the substitution y=vx and thereafter integrating the transformed equation.

)(vfdx

dvxv

dx

dy

Cvvf

dvx

)(ln

Page 52: Kuliah Diff Eq WS

First-order ordinary differential equation

)(sin)sin()ln()ln(sin

ln)ln(sinsin

cos

lncot

)tan(set

)tan( :Ex

1

12

1

AxxyAxx

yAx

x

y

cxcvdvv

v

cxx

dxvdv

vvdx

dvxv

dx

dyv

x

y

x

y

x

y

dx

dy

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QPydx

dy

where P and Q are functions of x only

Assuming the integrating factor R is a function of x only, then

dx

dRy

dx

dyRRy

dx

dRQyRP

dx

dyR )(

PdxR exp is the integrating factor

Equations Solved by Integrating Factor

There exists a factor by which the equation can be multiplied so that the one side becomes a complete differential equation. Thefactor is called “the integrating factor”.

PRdx

dR

RQdx

Ryd

)( dxxPdxxPdxxP dxxQy )()()( Cee)(e

Page 54: Kuliah Diff Eq WS

Linear first order ODEs:

The general form of a linear first order ODE is

Linear first order ODEs

).()( xqyxpdx

dy

First-order differential equations

Cdssqdttpdttpx

Cdttqtxy

dttqtxg

yf

xgyxxgdtx

yfdttqtydttptyfdttqytptC

dttpCxdxxpd

xpxxqyxpxdy

d

dx

d

dyxdxxqyxpxyxdx

dydxxqyxpxqyxpdx

dy

x sx

x

x

y

x xx

x

)()(exp)(exp)(

)()()(

)()()(

0)(

)()()()(

)()()()()()()()()(

)(exp)()()()()()()(

0)()()()(),( then ,factor ngintergrati thebe )(Let

0)()()()(

2

2

1

All linear first order ODEs have a formulated solution.

Page 55: Kuliah Diff Eq WS

2

322

2

2

23

5

1)(

.)ln2exp(2

exp)(

2 ,2

:1 Example

x

CxCdsss

xxy

xxdtt

x

xyxdx

dyxy

dx

dyx

x

x

.2

1)(

.exp)(

:2 Example

xxx ssx

xx

x

CeeCdseeexy

edtx

eydx

dy

.)1ln(1

1)(

.11

exp)(

1

:3 Example

CxxCdss

s

sxxy

xdt

tx

x

x

x

y

dx

dy

x

x

Cdssqdttpdttpxyxqyxp

dx

dy x sx)()(exp)(exp)()()(

Page 56: Kuliah Diff Eq WS

Solve

2

3exp

24 x

ydx

dyxy

Let z = 1/y3

4

3

ydy

dz

dx

dy

ydx

dz4

3

2

3exp33

2x

dx

dzxz

integral factor

2

3exp3exp

2xxdx

32

3exp

2

3exp3

22

x

dx

dzxxz

32

3exp

2

xz

dx

dCx

xz

3

2

3exp

2

Cxx

y

3

2

3exp

1 2

3

Example

Page 57: Kuliah Diff Eq WS

Example

0cossin ydyydxSolve yN

yM

cos

sin

x

N

y

M

integral factor = F(x))cos()sin( yF

xyF

y

dx

dFyyF coscos

dxF

dF

xeF

0cossin ydyydx 0cosesine ydyydx xx

dyy

F'dx

x

F'dF'

ye

x

F' x sin

yey

F' x cos

CyeF x sin'

Page 58: Kuliah Diff Eq WS

0),(),( dyyxNdxyxM)(xF

0),()(),()( dyyxNxFdxyxMxF

)()( FNx

FMy

dx

dFN

x

NF

y

MF

dx

dFN

x

N

y

MF

F

dFdx

x

N

y

M

N

1

)(function1

xx

N

y

M

N

Equations Solved by Integrating Factor

Page 59: Kuliah Diff Eq WS

0),(),( dyyxNdxyxM)(yG

0),()(),()( dyyxNyGdxyxMyG

)()( GNx

GMy

x

NG

dy

dGM

y

MG

dy

dGM

y

M

x

NG

G

dGdy

y

M

x

N

M

1

)(function1

yy

M

x

N

M

Equations Solved by Integrating Factor

Page 60: Kuliah Diff Eq WS

Example

0)63()6( 22 dyxxydxxyySolve

2

2

63

6

xxyN

xyyM

xyx

N

xyy

M

123

62

x

N

y

M

yxyy

xy

y

xyxy

y

M

x

N

M

1

)6(

6

xy6

)62()123(12

G

dGdy

y

1yyG )(

Page 61: Kuliah Diff Eq WS

0)63()6( 22 dyxxydxxyy 0)63()6( 22 dyxxyydxxyyy

dyy

F'dx

x

F'dF'

)6( xyyyx

F' 2

)(3)()6(' 223 yfyxxyyfdxxyyyF 2

dy

ydfyxxy

y

F' )(63 22

yxxyy

F' 2 263

C)(0)(

yfdy

ydfCyxxy 223 3

Example

Page 62: Kuliah Diff Eq WS

First-order ordinary differential equation

Bernoulli’s equation

ODElinear )()1()()1(

)()1()()1(

)()()1

(

1)1(

linear nonlinear onsubstituti a make

1or 1 where )()(

11

1

xQnvxPndx

dvv

yxQnyxPn

dx

dv

v

yxQyxP

dx

dv

n

y

dx

dv

n

y

dx

dy

dx

dyyn

dx

dv

yv

nnyxQyxPdx

dy

nn

n

nn

n

n

Page 63: Kuliah Diff Eq WS

343

33333

13

3

33

33

434

44341

43

6 is solution the

6)6(1

)1

()()()(

1)(

1}

3exp{})(exp{)(..

6)( ,3

)( with ODElinear 63

23

12

3

33let

2 :Ex

cxxy

ycxxdxxxx

dxxQxx

xv

xdx

xdxxPxFI

xxQx

xPxvxdx

dv

xx

y

dx

dvyx

x

y

dx

dvy

dx

dvy

dx

dy

dx

dyy

dx

dvyyv

yxx

y

dx

dy

First-order ordinary differential equation

Page 64: Kuliah Diff Eq WS

Bernoulli Equation yxRyxQyxP )()(')( 0)())()(( dyxPdxyxRyxQ

byxfyx,F )()( 0)()())()(()( dyxPyxfdxyxRyxQyxf bb

])()([])()()()([ b1 bb yxPxfx

yxRxfyxQxfy

bbbb yxP'xfyxPxf'yxRxfbyxQxfb )()()()()()()()()()1( 1

)()()()()()()1( xP'xfxPxf'xQxf b

dyy

F'dx

x

F'dF'

Page 65: Kuliah Diff Eq WS

Bernoulli Equation

)(

)()1(

)(

)(

)(

)(

xP

xQ

xP

xP'

xf

xf'

dxxP

xQxPxf

)(

)()1()(ln)(ln

dxxP

xQ

exP

xf )(

)()1(

)(

1)(

ye

xPyx,F

dxxP

xQ

)(

)()1(

)(

1)(

Page 66: Kuliah Diff Eq WS

Example

232' yxyxy Solve 2�,3)(�,2)(�,)( xRxxQxxP

xdxx

xdxxP

xQ

ex

ex

exP

xf 22

)21()(

)()1( 11

)(

1)(

221)( ye

xyx,F x

0])32[(1 222 xdydxyxyyex

x dyy

F'dx

x

F'dF'

)3

2( 12

xye

x

F' x

22

yey

F' x )()(dy 1222 xfyexfyeF' xx

dx

xdfye

x

F' x )(2 12

Page 67: Kuliah Diff Eq WS

Example

x

e

dx

xdf x23)(

dxx

exf

x

23

)(

dxx

eyeF'

xx

212 3

Cdxx

eye

xx

212 3

Page 68: Kuliah Diff Eq WS

Riccati Equation

)()()(' 2 xRyxQyxPy

)(1

)()(1

)()(1

)('2

2xR

zxSxQ

zxSxPz'

zxS

z

xSy1

)(

zxQ

zxSxP

zxP

xRxSxQxSxPz'z

xS

1)(

1)()(2

1)(�����������

)]()()()()([1

)('

2

22

)()()()()(')( 2 xRxSxQxSxPxS )(xSy

Page 69: Kuliah Diff Eq WS

zxQ

zxSxP

zxPz'

z

1)(

1)()(2

1)(

122

)())()()(2( xPzxQxSxPz'

dxxQxSxP

exf)]()()(2[

)(

actoregrating f�int

)(dx)()(

)(

1)(

xf

CxfxP

xfxz

Riccati Equation

Page 70: Kuliah Diff Eq WS

Example

xx eyyey 2'SolvexexS )(

ze

zxSy x 11)(

xx exRxQexP )(�,1)(�,)(

xdxdxeeeeexf

xx

33]1)(2[)(actoregrating f�int

xx Ceexf

CdxxfxP

xfxz 3

2

1

)()()(

)(

1)(

13

2

1)(

xxx Ceeexy

Page 71: Kuliah Diff Eq WS

First-order ordinary differential equation

Linear equations

solution the is )()()(

1)()()(

)()(])([)(

)()()()((1) Eq.

})(exp{)()()()(

ODEexact an is 0))()()(()(

))()()(()(

)1()()()()()(

)(factor gintegratin multiply )()(

dxxQxx

ydxxQxyx

xQxyxdx

d

dx

xdy

dx

dyxyxPx

dx

dyx

dxxPxxPxdx

xd

dxxQyxPxdyx

xPxQxdx

dyx

xQxyxPxdx

dyx

xxQyxPdx

dy

Page 72: Kuliah Diff Eq WS

First-order ordinary differential equation

)exp(2

)exp())(exp(2)2ln(

22

22

)2(2

method separated-Variable (2)

)exp(2

)exp(2)exp(4)exp(

)exp(}2exp{)( (1)

42 :Ex

2

222

2

222

2

xky

xkcxycxy

xdxy

dyxdx

y

dyyx

dx

dy

xcy

cxdxxxxy

xxdxx

xxydx

dy

Page 73: Kuliah Diff Eq WS

First-order ordinary differential equation

Isobaric equations

mvxydxxm

dyy

yxB

yxA

dx

dy

onsubstituti a make then , and to relative weight

a given each are and if consistent llydimensiona is equation The

),(

),(

cxxycxyxcxv

x

dxvdv

vxdx

dv

x

v

dx

dvvx

xx

v

vxdx

dvxvx

dx

dy

vdxxdvxdyvxym

yxdydxx

y

xy

yxdx

dy

ln2

1ln)(

2

1ln

2

1

122

)2

(2

1RHS

2

1LHS

2/2/1

lyrespective 1,2m 0, 1,2m is litydimensiona the 02)2

(

)2

(2

1 :Ex

222/12

21

2

2/12/12/3

2/32/12/1

2

2

Page 74: Kuliah Diff Eq WS

Miscellaneous equations

First-order ordinary differential equation

)(

onsubstituti a make

)( (1)

xbFadx

dyba

dx

dv

cbyaxv

cbyaxFdx

dy

11

11

2

2

2

)1(tan

tan1

111

)1( :Ex

cxyx

cxvdxv

dv

vdx

dy

dx

dvyxv

yxdx

dy

Page 75: Kuliah Diff Eq WS

ODE shomogeneou a

0)((

0)()(

shomogeneou is RHS and let (2)

fYeX

bYaX

dX

dY

gfefYeXgYfX e

cbabYaXcYbXa

YyXxgfyex

cbyax

dx

dy

First-order ordinary differential equation

2223

123

1

2

2

)32)(34()21

1)(1

1

14()1(

)3exp()2)(14()2ln(3

2)14ln(

3

1ln

23

2

143

4

472

42

42

472

42

52

42

52let

42

52

10642 and 0352

,let 642

352 :Ex

cxyxycx

y

x

yx

cvvXcvvX

dX

dX

v

dv

v

dvdv

vv

v

v

vv

dX

dvX

v

v

vXX

vXX

dX

dvXv

dx

dvXv

dX

dYvXY

YX

YX

dX

dY

YyXxyx

yx

dx

dy

Page 76: Kuliah Diff Eq WS

Second Order Differential Equations

Purpose: reduce to 1st O.D.E.

Likely to be reduced equations:

-Non-linear Equations where the dependent variable does not occur explicitly Equations where the independent variable does not occur explicitly Homogeneous equations

-Linear The coefficients in the equation are constant The coefficients are functions of the independent variable

)(2

2

xfdx

yd

Page 77: Kuliah Diff Eq WS

Second order Ordinary Differential Equation

77

Page 78: Kuliah Diff Eq WS

Higher-degree first-order equation

First-order ordinary differential equation

0),().....,(),(),( is solution general The

,...2,1for ),( equation of solution the is 0),(

),( and ),(

0))........()((

for 0),(),(...........),(

321

21

011

1

yxGyxGyxGyxG

niyxFdx

dypyxG

yxFpyxFF

FpFpFpdx

dypyxapyxapyxap

n

ii

iii

n

nn

n

0)]1()][1([ is solution general The

0)1()1ln(ln1

202)1( (2)

0)1(1lnln1

0)1( (1)

0]2)1][()1[(

for 02)123()1( :Ex

221

222

22

2

11

2

22223

xkyxky

xkycxyx

xdx

y

dyxy

dx

dyx

xkyc)(xyx

dx

y

dyy

dx

dyx

xypxypx

dx

dypxyypxxpxxx

Page 79: Kuliah Diff Eq WS

dy

dp

p

F

y

F

pdy

dx

dx

dyppyFx

x

1

for ),(

for soluable Equation

First-order ordinary differential equation

xyyxy

pxypyxpypy

ypyp

kkxyykxkyy

kx

y

ky

y

kpkpyc

ypcyp

dyyp

dpp

dy

dpy

dy

dpyp

dy

dpypypyp

dy

dpy

dy

dp

p

y

ppdy

dxpy

p

yx

dxdypyxppy

p

solutionsingular 038)6/1(94

9)16()3()6( equation. origional the Change

6/1061(2)

solution general 6303603)(6

1lnlnln2ln

2202 (1)

0)2)(61(12613

363

/for 036 :Ex

2322

22222222

22

23322

22

2

22

2

222

2

22

Page 80: Kuliah Diff Eq WS

First-order ordinary differential equation

dx

dp

p

F

x

Fp

dx

dypxFy

y

),(

for soluable Equation

solutionsingular 0021 (2)

solution general 4)(

4)2()(eq. origion the intoput

1lnln

202 (1)

0)2)(1(

0)1(2)1(0)1(2

2222

02 :Ex

2

222

2

2

22

2

yxyxxp

kxky

kxxpykkxp

cx

px

dx

p

dp

dx

dpxp

dx

dpxpp

pdx

dpxpppp

dx

dpxp

dx

dpxp

dx

dpxppp

dx

dyxpxpy

yxpxp

Page 81: Kuliah Diff Eq WS

First-order ordinary differential equation

Clairaut’s equation

ODE origional the in eliminate 0),(0 (2)

solution general )()(

)(eq. origional the intoput

0 (1)

0)(

1112

12211

212

2

ppxGxdp

dF

cFxcycFc

cFxccxccdx

dyp

cxcydx

yd

dx

dp

xdp

dF

dx

dp

dx

dp

dp

dF

dx

dpxpp

dx

dy

)( pFpxy

solutionsingular 04442

2

02 (2)

solution general ))(()( (1)

0)2(2

:Ex

2222

22

2

yxx

yxx

yx

ppx

ccxcFxyppF

pxdx

dpp

dx

dpp

dx

dpxp

dx

dy

ppxy

Page 82: Kuliah Diff Eq WS

Non-linear 2nd O.D.E.

They are solved by differentiation followed by the p substitution.

When the p substitution is made in this case, the second derivative of y is replaced by the first derivative of p thus eliminating y completely and producing a first O.D.E. in p and x.

dx

dyp

2

2

dx

yd

dx

dp

- Equations where the dependent variables does not occur explicitly

Page 83: Kuliah Diff Eq WS

Solve axdx

dyx

dx

yd

2

2

Let dx

dyp

2

2

dx

yd

dx

dpand therefore

axxpdx

dp

2

2

1exp xintegral factor

222

2

1

2

1

2

1xxx

axexpeedx

dp

22

2

1

2

1

)(xx

axepedx

d

Bx

Aerfaxy

dxxCaxy

2

2

1exp 2

error function

Example

dx

dyxCap )

2

1exp( 2

Page 84: Kuliah Diff Eq WS

Let dx

dyp

dy

dpp

dx

dy

dy

dp

dx

dp

dx

yd

2

2

Non-linear 2nd O.D.E.- Equations where the independent variables does not occur explicitly

They are solved by differentiation followed by the p substitution.

When the p substitution is made in this case, the second derivativeof y is replaced as

Page 85: Kuliah Diff Eq WS

Solve 22

2

)(1dx

dy

dx

ydy

Letdx

dyp and therefore

21 pdy

dpyp

Separating the variables

dy

dpp

dx

yd

2

2

dyy

dpp

p 1

12

)1ln(2

1lnln 2 pay

)1( 22 yadx

dyp

)sinh(

1

)(sinh1

)1(1

22

caxa

y

bayax

ya

dyx

Example

Page 86: Kuliah Diff Eq WS

x

yf

dx

dy

2

2

dx

ydx

n

nn

dx

ydx 1

dx

dy

x

yf

dx

ydx ,

2

2

Non-linear 2nd O.D.E.- Homogeneous equations

The homogeneous 1st O.D.E. was in the form:

The corresponding dimensionless group containing the 2nd differential coefficient is

In general, the dimensionless group containing the nth coefficient is

The second order homogenous differential equation can be expressed in a form analogous to

If in this form, called homogeneous 2nd ODE

Page 87: Kuliah Diff Eq WS

y=vxdx

dvxv

dx

dy

2

2

2

2

2dx

vdx

dx

dv

dx

yd

dx

dvxvvf

dx

vdx

dx

dvx ,2 12

22

dx

dvxvf

dx

vdx ,22

22

x=et or t=lnxdt

dv

xdx

dt

dt

dv

dx

dv 1

dt

dv

dx

dvx

2

2

22

2

22

2

11

11

11

dt

vd

xdt

dv

x����

dt

dv

dt

d

dx

dt

xdt

dv

x����

dt

dv

dx

d

xdt

dv

xdx

vd

dt

dv

dt

vd

dx

vdx

2

2

2

22

dt

dvvf

dt

dv

dt

vd,22

2

Non-linear 2nd O.D.E.

Page 88: Kuliah Diff Eq WS

Solve2

222

222

dx

dyxy

dx

ydyx

Dividing by 2xy

2

2

2

2

1

2

1

dx

dy

y

x

x

y

dx

ydx

homogeneous

2

2

2

2

dt

dv

dt

vdv

dx

dy

x

yf

dx

ydx ,

2

2

Let vxy 2

22

22 22

dx

dvx

dx

dvvx

dx

vdvx

Let tex

dt

dvp

22 pdv

dpvp

2)ln( CxBxy

Axy

Singular solution

General solution

Example

02 porp�� ��dv

dpv

Page 89: Kuliah Diff Eq WS

)(... 11

1

10 xyPdx

dyP

dx

ydP

dx

ydP nnn

n

n

n

where (x) is any function of x.

2nd Order Linear Differential Equations

They are frequently encountered in most chemical engineering fields of study, ranging from heat, mass, and momentum transfer toapplied chemical reaction kinetics.

The general linear differential equation of the nth order having constant coefficients may be written:

Page 90: Kuliah Diff Eq WS

)(2

2

xRydx

dyQ

dx

ydP

where P, Q and R are constant coefficientsLet the dependent variable y be replaced by the sum of the two new variables: y = u + vTherefore

)(2

2

2

2

xRvdx

dvQ

dx

vdPRu

dx

duQ

dx

udP

If v is a particular solution of the original differential equation

The general solution of the linear differential equation will be the sum of a “complementary function” and a “particular solution”.

purpose

2nd Order Linear Differential EquationsThe general equation can be expressed in the form

02

2

Ru

dx

duQ

dx

udP

Page 91: Kuliah Diff Eq WS

02

2

Rydx

dyQ

dx

ydP

Let the solution assumed to be:mx

meAy

mxmmeA

dx

dy mx

m emAdx

yd 22

2

0)( 2 RQmPmeA mxm

auxiliary equation (characteristic equation)

Unequal rootsEqual rootsReal rootsComplex roots

The Complementary Function

Page 92: Kuliah Diff Eq WS

xmeAy 11 xmeAy 2

2

xmxm eAeAy 2121

Unequal Roots to Auxiliary EquationLet the roots of the auxiliary equation be distinct and of values m1 and m2. Therefore, the solutions of the auxiliary equation are:

The most general solution will be

If m1 and m2 are complex, it is customary to replace the complex exponential functions with their equivalent trigonometric forms.

0652

2

ydx

dy

dx

yd0652 mm

3

2

2

1

m

mxx BeAey 32

Solve

Page 93: Kuliah Diff Eq WS

mxAey mxmx mVe

dx

dVe

dx

dymxVey Let mxmxmx Vem

dx

dVme

dx

Vde

dx

yd 22

2

2

2

2

where V is a function of x

02

2

Rydx

dyQ

dx

ydP

02

2

dx

VdDCxV

mxeDCxy )(

Equal Roots to Auxiliary EquationLet the roots of the auxiliary equation equal and of value m1 = m2 = m. Therefore, the solution of the auxiliary equation is:

02 RQmPm 02 QPm

Page 94: Kuliah Diff Eq WS

Solve096

2

2

ydx

dy

dx

yd

auxiliary equation

0962 mm

321 mm

xeBxAy 3)(

Example

Solve 0542

2

ydx

dy

dx

yd

auxiliary equation

0542 mm

im 2

xixi BeAey )2()2(

)sincos(2 xFxEey x

Page 95: Kuliah Diff Eq WS

)(2

2

xRydx

dyQ

dx

ydP

Particular IntegralsTwo methods will be introduced to obtain the particular solution of asecond order linear O.D.E.

The method of undetermined coefficients ~confined to linear equations with constant coefficients and particular form of (x)

The method of inverse operators ~general applicability

Page 96: Kuliah Diff Eq WS

When (x) is a polynomial of the form where all the coefficients are constants. The form of a particular integral is

)(2

2

xRydx

dyQ

dx

ydP

RCy /

nn xaxaxaa ...2

210

nn xxxy ...2

210

Method of Undetermined Coefficients

When (x) is constant, say C, a particular integral of equation is

Page 97: Kuliah Diff Eq WS

Example

Solve 32

2

8444 xxydx

dy

dx

yd

32 sxrxqxpy 232 sxrxq

dx

dy

sxrdx

yd62

2

2

3322 84)(4)32(4)62( xxsxrxqxpsxrxqsxr

equating coefficients of equal powers of x

84

0124

4486

0442

s

sr

qrs

pqr

32 26107 xxxy p

0442 mm

auxiliary equation

xc eBxAy 2)(

32 26107)( xxxeBxA��

yyy

x

pc

0442

2

ydx

dy

dx

yd

complementary function

2m

Page 98: Kuliah Diff Eq WS

Method of Undetermined Coefficients

rxey

When (x) is of the form Terx, where T and r are constants. Theform of a particular integral is

RQrPr

T

2

When (x) is of the form Gsinnx + Hcosnx, where G and H are constants, the form of a particular solution is

nxMnxLy cossin

2222

2

)(

)(

QnPnR

nQHGPnRL

2222

2

)(

)(

QnPnR

nQGHPnRM

Page 99: Kuliah Diff Eq WS

Example

Solve1863

2

2

dx

dy

dx

yd

Cxy C

dx

dy

02

2

dx

yd

18)C(6)0(3

3C

xy p 3

063 2 mmauxiliary equation

xc BeAy 2

x

pc

BeAx��

yyy

23

0632

2

dx

dy

dx

yd

complementary function

20 andm� ��m

Page 100: Kuliah Diff Eq WS

Example

Solve xeydx

dy

dx

yd 42

2

78103

xCxey 4xCex

dx

dy 4)41(

xCexdx

yd 42

2

)816(

71024 CC

2

1C

xp xey 4

2

1

0)4)(23(8103 2 mmmmauxiliary equation

xxc BeAey 43/2

xxx

pc

BeAexe��

yyy

43/24

2

1

081032

2

ydx

dy

dx

yd

complementary function

43/2 andm� ��m

Page 101: Kuliah Diff Eq WS

Example

Solvexy

dx

dy

dx

yd2cos526

2

2

xDxCy 2sin2cos

)2cos2sin(2 xDxCdx

dy

)2sin2cos(42

2

xDxCdx

yd

0102

52210

DC

DC

1

5

D

C

xxy p 2sin2cos5

0)3)(2(62 mmmmauxiliary equation

xxc BeAey 32

xxBeAe��

yyyxx

pc

2sin2cos532

062

2

ydx

dy

dx

yd

complementary function

32 andm� ��m

Page 102: Kuliah Diff Eq WS

Example

xydx

dy

dx

yd2cos526

2

2

ixeydx

dy

dx

yd 22

2

526

ixp key 2

ix

ix

kedx

yd

ikedx

dy

22

2

2

4

2

ixixixix ekeikeke 2222 52624

ixix ekei 22 52)210( 52)210( ki

ii

ii

i

ik

5

4100

)210(52

)210)(210(

)210(52

)210(

52

)2sin2)(cos5()5( 2 xixieiy ixp xxy p 2sin2cos5

取實數部分

Page 103: Kuliah Diff Eq WS

)(2

2

xRydx

dyQ

dx

ydP

Particular IntegralsTwo methods will be introduced to obtain the particular solution of asecond order linear O.D.E.

The method of undetermined coefficients ~confined to linear equations with constant coefficients and particular form of (x)

The method of inverse operators ~general applicability

Page 104: Kuliah Diff Eq WS

n

nn

dx

ydyD

dx

ydyDDyD

dx

dyDy

...

)(2

22

But, 22)(

dx

dyDy

ydx

dy

dx

yd23

2

2

yDDyDDyDyyD )2)(1()23(23 22

Method of Inverse OperatorsSometimes, it is convenient to refer to the symbol “D” as the differential operator:

Page 105: Kuliah Diff Eq WS

The differential operator D can be treated as an ordinary algebraicquantity with certain limitations.

(1) The distribution law:A(B+C) = AB + ACwhich applies to the differential operator DD(u+v+w)=Du+Dv+Dw

(2) The commutative law:AB = BAwhich does not in general apply to the differential operator DDxy xDy (D+1)(D+2)y = (D+2)(D+1)y

(3) The associative law:(AB)C = A(BC)which does not in general apply to the differential operator DD(Dy) = (DD)y D(xy) = (Dx)y + x(Dy)

Method of Inverse Operators

Page 106: Kuliah Diff Eq WS

pxpx

pxnpxn

pxpx

epfeDf

epeD

peDe

)()(

...

pxpx eppeDD )23()23( 22 ypDfeyeDf

ypDeyeD

ypDeyeD

ypDeyDeDyeyeD

pxpx

npxpxn

pxpx

pxpxpxpx

)())((

)()(

...

)()(

)()(22

More convenient!

Differential Operator Properties

pxpppxD

pxppxD

pxpppxD

pxppxD

pxipxe

nn

nn

nn

nn

ipx

sin)()(cos

cos)()(cos

cos)()(sin

sin)()(sin

sincos

212

22

212

22

ipxnipxnipxnn eipeDeDpxD )Im(ImIm)(sin

where “Im” represents the imaginary part of the function which follows it.

Page 107: Kuliah Diff Eq WS

The operator D signifies differentiation, i.e.

)()( xfdxxfD )()( 1 xfDdxxf

D-1 is the “inverse operator” and is an “integrating” operator.It can be treated as an algebraic quantity in exactly the same manner as D

Inverse Operator

Page 108: Kuliah Diff Eq WS

Solvexey

dx

dy 24 differential operator

xeyD 2)4(

xeD

y 2

)4(

1

xeDDDy 232 ...])4

1()

4

1()

4

1(1[

4

1

binomial expansion

xey 2

2

1

pxpx epfeDf )()(

x2e)D

41

1(4

1y

2pxey 2

)42(

1

xx eey 2322

2

1...])

2

1()

2

1()

2

1(1[

4

1

Example

Page 109: Kuliah Diff Eq WS

Solve xxeydx

dy

dx

yd 42

2

6168

differential operator

xxeyDyDD 422 6)4()168( 01682 mm

xc eBxAy 4)(

xp xe

Dy 4

2)4(

6

xDey xp

246

f(p) = 0

)()( pDfeeDf pxpx

integration

xxp exxDey 432143

Example

01682

2

ydx

dy

dx

yd

4m

x

pc

exBxA��

yyy

43)(

Page 110: Kuliah Diff Eq WS

xxeydx

dy

dx

yd 42

2

6168

xeCxy 42xx eCxCxe

dx

dy 424 42

xxx eCxCxeCedx

yd 42442

2

16162

xx xeCe 44 62

xeCxy 43 xx eCxeCxdx

dy 4342 43

xxx eCxeCxCxedx

yd 434242

2

16246

xx xeCxe 44 66

1C

xp exy 43

Example

Page 111: Kuliah Diff Eq WS

Solve 232

2

346 xxydx

dy

dx

yd

differential operator232 34)2)(3()6( xxyDDyDD 062 mm

xxc BeAey 23

)34()2)(3(

1 23 xxDD

y p

expanding each term by binomial theorem

)34()2(

1

)3(

1

5

1 23 xxDD

y p

)34(...16842

1...

812793

1

5

1 233232

xxDDDDDD

y p

...01296

2413

216

)624(7

36

612

6

34 223

xxxxx

y p

Example

062

2

ydx

dy

dx

yd

23 and� ��m

108/)5661872( 23

23

xxx����

BeAe��

yyy

xx

pc

Page 112: Kuliah Diff Eq WS

232

2

346 xxydx

dy

dx

yd

32 sxrxqxpy

232 sxrxqdx

dy

sxrdx

yd62

2

2

062

0626

363

46

pqr

qrs

rs

s

108/)5661872( 23 xxxy p

Example

equating coefficients of equal powers of x

Page 113: Kuliah Diff Eq WS

nth – order linear differential equation

1. nth – order linear differential equation with constant coefficients.

xgyadx

dya

dx

yda

dx

yda

dx

yda

n

n

nn

n

n

012

2

21

1

1 ....

2. nth – order linear differential equation with variable coefficients

xgyxadx

dyxa

dx

ydxa

dx

ydxa

dx

dyxa

n

n

nn

012

2

2

1

1 ......

113

Page 114: Kuliah Diff Eq WS

A linear ODE of order n is

Higher-order ordinary differential equation

ousinhomogene is otherwise s;homogeneou is ODE the then 0)( If

)()()(.....)()( 011

1

1

xf

xfyxadx

dyxa

dx

ydxa

dx

ydxa

n

n

nn

n

n

0...... and

0)(......)()()(

,........,, constantsexist must

there interval, anover t independen linearly be to functions nFor

)(.....)()()(

so it, satisfy functionst independen linearly n Find

0)()(.......)()(

0)( with )( equation arycomplementFor 1.

321

332211

321

2211

011

1

1

cccc

xycxycxycxyc

cccc

xycxycxycxy

yxadx

dyxa

dx

ydxa

dx

ydxa

xfxy

n

nn

n

nnc

n

n

nn

n

n

c

Page 115: Kuliah Diff Eq WS

dependent. linearly are functions thethat guaranteenot does 0 :Note*

functions ofset the of Wronskianthe called is

0

......

............

...

...

),...,(

that sucht independen linearly

are )( functions n the ,0.......for

0)(....)()(

. . . . .

0)(.......)()(

0)(.......)()(

are statements equal An

)1()1(1

''2

'1

21

21

321

)1()1(22

)1(11

''22

'11

2211

W

W

yy

yyy

yyy

yyyW

xycccc

xycxycxyc

xycxycxyc

xycxycxyc

nn

n

n

n

n

in

nnn

nn

nn

nn

Higher-order ordinary differential equation

)()(y(x) solution Genaral

0)( with )( integral particularFor 2.

xyxy

xfxy

pc

p

Page 116: Kuliah Diff Eq WS

)(........ 011

1

1 xfyadx

dya

dx

yda

dx

yda

n

n

nn

n

n

Higher-order ordinary differential equation

Linear equations with constant coefficients

constants arbitrary are and )cos( or

)sin(

)sincos(

root a also is conjugatecomplex its then , isroot one If

complex roots Some (ii)

.......)(

distinct and real roots All(i)

0........

equation auxiliary obtain to )( solution trial theSet

0.........

from )( function arycomplemenr the Finding

21)(

2)(

1

21

011

1

011

1

1

21

AxAe

xAe

xdxdeece c

ii

ecececxy

aaaa

Aexy

yadx

dya

dx

yda

dx

yda

xy

x

x

xxixi

xn

xxc

nn

nn

x

n

n

nn

n

n

c

n

Page 117: Kuliah Diff Eq WS

xn

xlk

xllkkk

xkkc

xn

xk

xkkc

xxkxx

nlk

nk

ecec

excxcc

excxccxy

l-fold k-fold

ececexcxccxy

eexexxe

k

...........

).......(

)......()(

root a is root, a is If

...).....()(

except solutions, also are ......... , ,

that find we onsubstitutidirect by times, occurrs If

repeated roots Some (iii)

1

2

1

11

1111

1

121

121

21

11

21

12

1

15 Higher-order ordinary differential equation

Ex: Find the complementary function of the equation

xeydx

dy

dx

yd 2

2

2

xc

x exccyAey

ydx

dy

dx

yd

)(1012 try

02

212

2

2

Page 118: Kuliah Diff Eq WS

. )( tosimilar

from zedparameteri a assuming byor inspection by found be often can

)( ,)( simple a ans tscoefficienconstant with ODElinear For

)( integral particular the Finding

xf

xyxf

xy

p

p

Higher-order ordinary differential equation

(method of undetermined coefficients)

function. trial individual ingcorrespond the ofproduct or sum the as )(

try then above, the of any ofproduct or sum the the is )( If (iv)

......)( try then

zero) be may (some .....)( If (iii)

cossin)( try

then zero) be may or ( cossin)( If (ii)

)( try then )( If (i)

follows as are functions trial Standard

10

10

21

2121

xy

xf

xbxbbxy

axaxaaxf

rxbrxbxy

aarxarxaxf

bexyaexf

p

NNp

mN

N

p

rxp

rx

. ofpower integer smallest the by multiplied be should function trial the ),(

function arycomplement the within contained also is function trial the If :Note

xxyc

Page 119: Kuliah Diff Eq WS

Ex: Find the particular integral of the equation

Higher-order ordinary differential equation

xeydx

dy

dx

yd 2

2

2

xp

xxxxxx

xp

xc

xp

exxy

bb

bbbxbbbx

eebxexxebexxedx

db

ebxxy

exccxybexy

2

2

222

2

21

2

1)(

2

112

12)44()2(

)2(2)2(

)( Try

)()(but ,)(Set

Page 120: Kuliah Diff Eq WS

xxxxxxxdxdxyxyy

xxxxxxxy

bdecbe/bddc bebdaead

cbxaxdbaxex

ecbdbeadae

xcbxaxebdadxx

xxxexdcbxaxxexdbax

xxyyyyyyxxyyyydx

d

xyxyxeexdcxbxaxxy

xdxdececxyi

Aeyydx

yd

xxydx

yd

pc

p

p

ixixc

x

32/2cos16/2sin12/2cos2sin2cos)()(

32/2cos16/2sin12/2cos)(

32/12/ 0 161 2 3/4 0

0)23(4)26( : term 2cos thefor

2 86 12/1

)23()4(26 :term 2sin thefor

2sin)2sin22cos2)(23(2)2cos2sin)(26(

2sin204for 2sin)(4)(

)()()2cossin)(()(set (ii)

2sin2cos)(204

solution trial the 04for )i(

2sin4 Solve :Ex

2321

23

2

22

22

2'2

'12

''12

''2

221212

2

2123

212

22

12

2

2

22

2

Higher-order ordinary differential equation

Page 121: Kuliah Diff Eq WS

Laplace transform method

xxx

x

nnnnnn

sx

eeexy

ssssss

sssy

sssyss

ssyysysysysysSol

eydx

dy

dx

yd

fsffsfssfssf

dxexfsf

2

2

2

'2

2

2

)1()2('21)(

0

3

12

3

1)(

)2(3

1

1

2

)1(3

1

)2)(1)(1(

332)(

1

252)()23(

1

2)(2)]0()([3)0()0()( :

223 :Ex

)0()0().....0()0()()(

)()(

Higher-order ordinary differential equation

Page 122: Kuliah Diff Eq WS

15 Higher-order ordinary differential equation

Linear equation with variable coefficients

Legendre’s linear equation:

)(...)1)...(2)(1(

)1).....(2)(1()(

)1.().........2)(1()(

)23(]}[{

)1()(

][)()()(

Let

)()(.......)(

01

3

3

2

2

3

333

2

222

3

3

2

2

2

22

2

2

01

tn

n

n

n

n

n

tt

ttt

tt

n

nn

n

efya

dt

dyayn

dt

d

dt

d

dt

d

dt

da

yndt

d

dt

d

dt

d

dt

d

xdx

yd

yndt

d

dt

d

dt

d

dt

d

x

dt

dy

dt

yd

dt

yde

dt

dy

dt

yde

dt

d

dx

dt

dx

yd

ydt

d

dt

d

xdt

dy

dt

yde

dt

dye

dt

d

dx

dt

dt

dye

dx

d

dx

yd

dt

dye

dt

dy

xdt

dy

dx

dt

dx

dyex

xfyadx

dyxa

dx

ydxa

Page 123: Kuliah Diff Eq WS

Euler’s equation for

)(......... 01 xfyadx

dyxa

dx

ydxa

n

nn

n

0 and 1

22

21

22

21

22

2

2

222

2

2

2

2

2

22

)()(

204 try 04

)()()(

let :Sol

04 :Ex

xcxcxyececty

eyydt

yd

dt

yde

dt

dye

dt

yde

dt

dyee

dt

dye

dt

d

dx

dt

dt

dye

dx

d

dx

yd

dt

dye

dx

dt

dt

dy

dx

dyex

ydx

dyx

dx

ydx

tt

t

tt

ttttt

tt

Higher-order ordinary differential equation

Page 124: Kuliah Diff Eq WS

Higher-order ordinary differential equation

exact

'00

0'11

)1(001

'22

)1(0

)2(1

)1(1

)2(1

)3(2

)2(2

)5(5

)4(4

)3(3

)2(2

)1(10

)3(2

)4(3

)3(3

5n

3'

22

)4(3

)5(4

)4(42

'11

)5(4

)5(51

011

1

101

01

......

......

0

])(.....)([)()(..........)(

if , is )()()(.........)(

ODElinear order th-n The

ba

bba

babba

bba

bba

aaaaaabbabba

bbabba

baba

ybdx

dyxb

dx

ydxb

dx

dyxa

dx

dyxa

dx

ydxa

xfyxadx

dyxa

dx

ydxa

nnn

nnn

nn

n

n

nn

n

n

n

n

n

Page 125: Kuliah Diff Eq WS

Higher-order ordinary differential equation

0)()1(

)()1(.......)()()(

)(

0

)(''2

'10

xa

xaxaxaxa

nn

n

n

nn

n

Ex:13)1(

2

22 y

dx

dyx

dx

ydx

ODElinear order first 1)1(

)1( 1])1[(

1 1])()([

0)()1(1 3 1

21

2

122

0'02

2

1'101

)(2

001

22

x

cxy

x

x

dx

dy

cxxydx

dyxxy

dx

dyx

dx

d

dx

dybyb

dx

ydb

dx

dybyxb

dx

dyxb

dx

d

xaaxaxa nn

n

n

Page 126: Kuliah Diff Eq WS

Higher-order ordinary differential equation

1)1(

sin)(

sin)1(

)1(

1

)1(

)1(1

)()1()()1(

)()()()(

1}1

exp{})(exp{)(factor gintegratin

1)(

1)( )()(

2/122

11

21

12/12

2/1212/12

2/121

21

2/122/12

22

21

2

x

cxcxy

cxcx

dxx

cdxx

x

dxx

cxdx

x

cxxxyx

dxxQxxyx

xdxx

xdxxPx

x

cxxQ

x

xxPxQyxP

dx

dy

Page 127: Kuliah Diff Eq WS

Higher-order ordinary differential equation

Partially known complementary function

.in 1-norder of equation an into above the

transform can we ,)()()( onsubstituti the making

function, arycomplement the ofpart one is )( ,0)( As

)()()(.........)(

ODElinear order -nth

01

dx

dv

xvxuxy

xuxf

xfyxadx

dyxa

dx

ydxa

n

n

x

x

dx

dvx

dx

vdx

dx

dvx

dx

vdx

x xxvxxvxxvdx

dxxvxvxuxy

xxxuxcxcxy

xydx

yd

c

cos

csctan2 cscsin2cos

csccos)()sin)(cos)(()(cos)()()(

)sin(or cos)(let cossin)(

csc :Ex

2

2

2

2

'

21

2

2

Page 128: Kuliah Diff Eq WS

Higher-order ordinary differential equation

xxxxxcxc

xxvxy

cxcxxxxv

xxdx

xxxxdxx

xxxdxx

xvxvxxuxu

xdxcxdxxxv

cxxdxdx

dvxx

dx

dvx

dx

d

xdx

dvxx

dx

vdx

xxscexxdxx

cossinlnsincossin

)(cos)( solution general

tansinlntan)(

tansec and

sinlntantansin

cossinlntansinlnsec

tan sec sin/cos sinln part by integral

secsinlnsec)(

sinlncotcos cot)(cos

ODElinear exact a is cotcossin2cos

cos}cosexp{ln}ln2exp{}tan2exp{)(

21

21

2

2

2''

21

2

122

2

22

22

Page 129: Kuliah Diff Eq WS

Higher-order ordinary differential equation

Variation of parameters

)......(.......

)....(.....

nmFor

)......(.......)(

0...... :constraint the choose we **

).......(......)(

)()(......)()()()()(

is )( integral particular a Assume

known are ,...1 )( all )(...........)()()(

,)( function arycomplement the ,0)(For

)()(.........)(

)1(')1(1

'1

)()(11

)(

)1(')1(1

'1

)()(11

)(

'''1

'1

''''11

''

'1

'1

'1

'1

''11

'

2211

2211

01

nnn

nnnn

nnp

mnn

mmnn

mmp

nnnnp

nn

nnnnp

nnp

p

innc

c

n

n

n

ykykykyky

ykykykyky

ykykykykxy

ykyk

ykykykykxy

xyxkxyxkxyxkxy

xy

nixyxycxycxycxy

xyxf

xfyadx

dyxa

dx

ydxa

Assumed to be zero

Not zero

Page 130: Kuliah Diff Eq WS

Higher-order ordinary differential equation

)()........(

0........ :)( solution arycomplementFor

)()....()......(

)()....(

)().....().....(

)1(')1(1

'1

0'

1)(

)1(')1(1

'10

'1

)(

1

)1(')1(1

'1

)(

10

)1(')1(1

'1

)()(11

0

xfykyka

yayayaxy

xfykykayayayak

xfykykayka

xfykykaykyka

nnn

nn

jjn

jnj

nnn

nnjj

njn

n

jj

nnn

nn

mj

n

jj

n

mm

nnn

nn

mnn

mn

mm

)(/)()()(......)()(

0)()(.....)()(

....

....

0)()(..............)()(

0)()(.............)()(

)1(')1(1

'1

)2(')2(1

'1

'''1

'1

'1

'1

xaxfxyxkxyxk

xyxkxyxk

xyxkxyxk

xyxkxyxk

nn

nnn

nnn

n

nn

nn

n constraints to be used to solve

)(xkn

)()]([

)()()(

1

xyxkc

xyxyxy

mm

n

mm

pc

Page 131: Kuliah Diff Eq WS

0)2

()0( csc:Ex2

2

yyxydx

yd

xxxxxy

ccyy

xxcxxcxy

xxkxxxk

xxkxxxxk

xxxkxxk

xxkxxk

xxkxxkxy

xcxcxy

p

c

cossin)sin(ln)(

0 0)2

()0(

cos)(sin)sinln()(

)( 1cscsin)(

sinln)( cotsin/cos)(

cscsin)(cos)(

0cos)(sin)(

:sConstraint

cos)(sin)()(

cossin)(

21

21

2'2

1'1

'2

'1

'2

'1

21

21

Higher-order ordinary differential equation

Page 132: Kuliah Diff Eq WS

Higher-order ordinary differential equation

Green’s functions

z

zm

mn

m

z

z m

b

a

b

a

n

n

n

dxzxdxdx

zxGdxa

zx

zxG

zbGzaGbyay

zxG

zxzxLG

xfdzzfzxLGxLy

dzzfzxGxy

bxa

zxG

xfxLy

xfyxadx

dyxa

dx

ydxa

1)(lim),(

)(lim

at sderivative its

and ),( of itydiscontinuor continuity The II)(

0),(),( 0)()( if

conditions boundary the obeys ),( (I)

)(),(

)()()],([)(

)(),()(

. range the in

conditions boundary some obey and exists ),( function a Suppose

)()( equationoperator as written be can

)()()(........)(

000

01

Page 133: Kuliah Diff Eq WS

Higher-order ordinary differential equation

)(

1|

),(

1}),(

)(|),(

)({lim

),()(lim

For

0),(

)(lim

1 to 0For

point thisat continuous

bemust 1for derivativeorder -lower the all (3)

at itydiscontinu finite a havemust derivativeorder -1)th-(n (2)

infinite is value itsbut exists, )( (1)

1

1

1

1'

1

1

0

0

0

xadx

zxGd

dxdx

zxGdxa

dx

zxGdxa

dxdx

zxGdxa

nm

dxdx

zxGdxa

nm

nmdx

Gd

zx

zxdx

Gd

n

zzn

n

n

n

nzzn

n

n

n

nz

z n

m

mz

z m

m

m

n

n

Page 134: Kuliah Diff Eq WS

Higher-order ordinary differential equation

0)2

()0( csc :Ex2

2

yyxydx

yd

zxxz

zxxzzxG

zzDzzAxzAzzD

zzDzzA

zadx

dGzxzxG

zxxzD

zxxzAzxG

zCzB

zGzG

zxxzDxzC

zxxzBxzAzxG

xzzxGdx

zxGd

zz

for cossin

for sincos),(

sin)( cos)( 1cos)(sin)(

0cos)(sin)(

1)(

1| and at continuous is ),( (2)

for cos)(

for sin)(),(

0)()(

0),2/(),0( conditions boundary (1)

for cos)(sin)(

for cos)(sin)(),( Assume

)(),(),(

2

2

2

Page 135: Kuliah Diff Eq WS

Higher-order ordinary differential equation

xxp

x

x

dzzzfxdzzzfxxy

xfydx

yd

xxxx

zdzzxzdzzx

zdzzxGxy

)(cossin)(sincos)(

)( :Ex

integral particular a

findingfor usefyl also is method function sGreen' :Note

sinlnsincos

csccossincscsincos

csc),()(

:solution general

2

2

2/

0

2/

0

Page 136: Kuliah Diff Eq WS

Higher-order ordinary differential equation

0)0()0(for )( :Ex '2

2

yyxfydx

yd

x xdzzfzxdzzfzxGxy

zxxzxz

zxzxG

zzDzzC

zzDzzC

zzDzzCzx

zxxzDxzC

zxzxG

zBzAzGzG

zxxzDxzC

zxxzBxzAzxG

0 0

'

)()sin()(),()(

for cossinsincos

for 0),(

sin)( cos)(

1sin)(cos)(

0cos)(sin)( at

for cos)(sin)(

for 0),(

0)()( 0),0(),0(

for cos)(sin)(

for cos)(sin)(),(

Page 137: Kuliah Diff Eq WS

Higher-order ordinary differential equation

0)0()0( and 0)0(

),0(point the thriugh )(set

)0()0( (2)

and

0)()( and 0)()(

)(set

)( ,)( (1)

caseorder -second theconsider :exampleFor

0)0()0(or 0)()(

shomogeneou are conditions boundary the , variable

new the inthat such variable, of change a Make

)0()0(or )( ,)(

condition boundary ousinhomogenefor :Note

''

'

'

'

yuu

cm

cmxyu

yy

ba

bac

bam

cmbbucmaau

cmxyu

byay

uubuau

u

yybyay

Page 138: Kuliah Diff Eq WS

Higher-order ordinary differential equation

Canonical form for second-order equation

})(2

1exp{)()(/)()(

)(2

1)]([

4

1)()(

4

1

2

1

22

1 )(

2

1 and

2

1

)()(})(2

1exp{)( 0

2

)( in term no was there If **

)()2

( )()()(Let

)()()(

1

'1

210

0'

1''

21

'1

'1'

1

'''

1'1

''1

'

2

2

11

'

'

0'

1''

'1

'''

012

2

dzzaxfxuxfxh

xaxaxau

uauauxg

aau

uaa

u

uuauaua

u

u

xhvxgdx

vddzzaxua

u

u

xv

u

fv

u

uauauva

u

uvxvxuxy

xfyxadx

dyxa

dx

yd

Page 139: Kuliah Diff Eq WS

Higher-order ordinary differential equation

0)1(44 :Ex 22

22 yx

dx

dyx

dx

ydx

xxcxcxy

xcxcxv

xvdx

xvd

xxxxg

xdz

zxu

x

xxa

xxa

xvxuxyyx

x

dx

dy

xdx

yd

/)2

1cos

2

1sin()(

2

1cos

2

1sin)(

0)(4

1)(

4

1

2

1

4

1

4

1

4

1)(

1}

1

2

1exp{)(

4

1)(

1)(

)()()(set 04

11

21

21

2

2

222

2

2

01

2

2

2

2

Page 140: Kuliah Diff Eq WS

Higher-order ordinary differential equation

General ordinary differential equations

(I) Dependent variable absent : An ODE does not contain variables y, but only dy/dx, change variable p=dy/dx

222

1

21

122

1222

2

2

2

)(

12)(

2

2

1

2

1[44)()(

)2exp()(

42 Let :

42 :Ex

cxxecxy

dx

dyecxxp

cexe

cdxexedxxexpx

edxx

xpdx

dp

dx

dypSol

xdx

dy

dx

yd

x

x

xx

xxx

x

Page 141: Kuliah Diff Eq WS

Higher-order ordinary differential equation

(II) Independent variable absent: An ODE does not contain independent

variable x, except in d/dx, d2/dx2, making a substitution dy/dx=p

22

22

3

3

2

2

)()()(dy

dpp

dy

pdp

dy

dpp

dy

d

dx

dy

dy

dpp

dx

d

dx

yd

dy

dpp

dy

dp

dx

dy

dx

dp

dx

yd

2222/122

22

2/12

22

222122

222

2

22

2

)( )(-

)(

)1( )ln(1

1ln

1

)1( 01 Let

0)(1 :Ex

cyAxAxyc

dxdyyc

y

y

yc

dx

dyp

cypycpy

dydp

p

p-

pdy

dpypp

dy

dpyp

dy

dpp

dx

yd

dx

dyp

dx

dy

dx

ydy

Page 142: Kuliah Diff Eq WS

Higher-order ordinary differential equation

xdx

yd

dx

dy

dx

ydy

2

2

3

3

62 :Ex

Non-linear exact equation

3221

42

21

3

1

2

1

22

2

2

22

2

2

22

2

2

3

3

2

2

224

62

2)2(

2)(22

])(2[)2( )2()1(

4])(2[ (2)

22)2( (1)

cxcxcx

ycxcx

dx

dyy

cx

dx

dyy

dx

d

cx

dx

dy

dx

ydy

xdx

dy

dx

d

dx

ydy

dx

d

dx

yd

dx

dy

dx

dy

dx

d

dx

yd

dx

dy

dx

ydy

dx

ydy

dx

d

Page 143: Kuliah Diff Eq WS

Higher-order ordinary differential equation

Isobaric or homogeneous equations

)(),(:/(c) )1((b) )((a) :Weight

(2) (1)

nWxmW ydxydWxmWy

exvxynn

tm

0 and Let

equation shomogeneou 0)1(

2 set 1m weihght equalfor

1m 2m, 2m, 1,m 1,m

1m 2m, 2m, m),1-(2 m),2-(3 :Weight

0

0)()( :Ex

2

22

2

22

2

2

2

2

2

2

2

2

222

23

222

23

dt

dvv

dt

vd

dt

dve

dt

vde

dx

vd

dt

dve

dx

dvex

dx

dvv

dx

vdx

dx

dv

dx

vdx

dx

yd

dx

dvxv

dx

dyvxy

xyydx

dyxy

dx

dyx

dx

ydx

xyydx

dyxyx

dx

ydx

tttt

Page 144: Kuliah Diff Eq WS

Higher-order ordinary differential equation

)ln2

1tan(

)(tanln2

lnfor

)(tan1

2

2

1

)(2

1

2

1

2)

2(

2111

1

121

1

1

1

122

12

21

21

2

1

22

2

2

ddxdxdy

xd

yddx

dxt

d

v

dd

t

dv

dvdt

dvcvdt

dv

cv

dtv

dt

d

dt

dvdt

dt

dvvdt

dt

vd

Page 145: Kuliah Diff Eq WS

Higher-order ordinary differential equation

Equation homogeneous in x or y alone

221

21

221

2

21

21122

333

2

2

32

2

2

22

2

2

32

22

ln2

1

)1(2

)1

(2

)1

(2 1

2

1

2

2 0

2

Set 02

)( and Let

in shomogeneou 02

:Ex

cxctc

ycct

yc

ydyct

yc

dy

yc

dt

dypc

yp

dyy

pdpydy

dpp

ydy

dpp

dy

dpp

dy

dp

dt

dy

dt

dp

dt

yd

dt

dyp

ydt

yd

dt

dy

dt

yde

dx

yd

dt

dye

dt

dy

dx

dt

dx

dydtedxex

xydx

dyx

dx

ydx

tttt

Page 146: Kuliah Diff Eq WS

Higher-order ordinary differential equation

Equations having as a solution:

If any general nth-order ODE is satisfied identically by assuming

xAey

solution a is )exp( then /......// 22 xAydxyddxyddxdyy nn

solution a is y.identicall satisfied is Which

0)( Set

0)()( :Ex

222222

2

222

22

xAey

xyyxyxxdx

yd

dx

dyy

dx

dyx

dx

dyyx

dx

yd

dx

dyxx

Page 147: Kuliah Diff Eq WS

Simultaneous Diffusion and Reaction

A tubular reactor of length L and 1 m2 in cross section is employed to carry out a first order chemical reaction in which a material A isconverted to a product B,

A B

The specific reaction rate constant is k s-1. If the feed rate is u m3/s, the feed concentration of A is Co, and the diffusivity of A is assumed to be constant at D m2/s. Determine the concentration of A as a function of length along the reactor. It is assumed that there is no volume change during the reaction, and that steady state conditions are established.

Page 148: Kuliah Diff Eq WS

A material balance can be taken over the element of length x at a distance x from the inlet

The concentration will vary in the entry sectiondue to diffusion, but will not vary in the sectionfollowing the reactor. (Wehner and Wilhelm, 1956)

x x+x

Bulk flow of A

Diffusion of A

uC

xdx

dCD

dx

d

dx

dCD

dx

dCD

xdx

dCuuC

Input - Output -Reaction = Accumulation

0

xkCx

dx

dCD

dx

d

dx

dCDx

dx

dCuuC

dx

dCDuC

分開兩個 section

uC0

x

L

x

uC

C

Simultaneous Diffusion and Reaction

Page 149: Kuliah Diff Eq WS

0

xkCx

dx

dCD

dx

d

dx

dCDx

dx

dCuuC

dx

dCDuC

dividing by x

0

kC

dx

dCD

dx

d

dx

dCu

rearranging

02

2

kCdx

dCu

dx

CdD

auxiliary equation

02 kumDm

a

D

uxBa

D

uxAC 1

2exp1

2exp

In the entry section0

''2

2

dx

dCu

dx

CdD

02 umDm

D

uxC exp'

2/41 ukDa

Simultaneous Diffusion and Reaction

auxiliary equation

Page 150: Kuliah Diff Eq WS

a

D

uxBa

D

uxAC 1

2exp1

2exp

D

uxC exp'

B. C.

0

'0

dx

dCLx

dx

dC

dx

dCx

B. C.

CCx

CCx

'0

' 0

Simultaneous Diffusion and Reaction

012

exp)1(12

exp)1(

)1()1(2

0

aD

uxaBa

D

uxaA

aBaA

BA

C

)1()1(2 0 aBaAC

Page 151: Kuliah Diff Eq WS

xL

D

uaaxL

D

uaa

D

ux

KC

C

2exp1

2exp1

2exp

2

0

if diffusion is neglected (D0)

u

kx

C

CCexp1

0

0

DuLaaDuLaaK 2/exp12/exp1 22

D

uLa

K

aCB

D

uLa

K

aCA

2exp

)1(2

2exp

)1(2

0

0

Simultaneous Diffusion and Reaction

Page 152: Kuliah Diff Eq WS

1.017 kg/s of a tallow fat mixed with 0.286 kg/s of high pressure hot water is fed into the base of a spray column operated at a temperature 232 C and a pressure of 4.14 MN/m2. 0.519kg/s of water at the same temperature and pressure is sprayed into the top of the column and descends in the form of droplets through the rising fat phase. Glycerine is generated in the fat phase by the hydrolysis reaction and is extracted by the descending water so that 0.701 kg/s of final extract containing 12.16% glycerine is withdrawn continuously from the column base. Simultaneously 1.121 kg/s of fatty acid raffinate containing 0.24% glycerine leaves the top of the column. If the effective height of the column is 22 m and the diameter 0.66 m, the glycerine equivalent in the entering tallow 8.53% and the distribution ratio of glycerine between the water and the fat phase at the column temperature and pressure is 10.32, estimate the concentration of glycerine in each phase as a function of column height. Also find out what fraction of the tower height is required principally for the chemical reaction. The hydrolysis reaction is pseudo first order and the specificreaction rate constant is 0.0028 s-1.

Continuous Hydrolysis of Tallow

Page 153: Kuliah Diff Eq WS

tallow fat hot water G’’ kg/s

glycerine (extract)

fatty acid (raffinate)L kg/s

L kg/sxH

zH

x0

z0

y0

G kg/syH

h

xz

x+xz+z y+y

y

h

Continuous Hydrolysis of Tallow

hot water G’ kg/s

Page 154: Kuliah Diff Eq WS

Consider the changes occurring in the element of column of height h:

glycerine transferred from fat to water phase, hyyKaS )*(

S: sectional area of towera: interfacial area per volume of towerK: overall mass transfer coefficient

rate of destruction of fat by hydrolysis, hSzk rate of production of glycerine by hydrolysis, whSzk /

k: specific reaction rate constant: mass of fat per unit volume of column (730 kg/m3)w: kg fat per kg glycerine

x = weight fraction of glycerine in raffinatey = weight fraction of glycerine in extracty*= weight fraction of glycerine in extract in equilibrium with xz = weight fraction of hydrolysable fat in raffinate

Continuous Hydrolysis of Tallow

Page 155: Kuliah Diff Eq WS

A glycerine balance over the element h is:

hyyKaSw

hSzkh

dh

dxxLLx

*

A glycerine balance between the element and the base of the tower is:

00 Gy

w

LzLxGy

w

Lz

L kg/sxH

zH

x0

z0

y0

G kg/syH

h

xz

x+xz+z y+y

y

h

The glycerine equilibrium between the phases is:

mxy *

hyyKaSGyhdh

dyyG *

in the fat phase

in the extract phase

Continuous Hydrolysis of Tallow

Page 156: Kuliah Diff Eq WS

hyyKaSGyhdh

dyyG *

dh

dyGKaSyKaSmx

00 Gy

w

LzLxGy

w

Lz

hyyKaSw

hSzkh

dh

dxxLLx

*

hyyKaSGyhdh

dyyG *

dh

dyG

dh

dxL

w

Szk

dh

dyG

dh

dxLxyy

L

G

w

zSk

0

0multiply by KaSm/LG

xyyL

G

w

z

w

z )( 0

0

02

2

00

2

dh

dy

L

KaSm

dh

yd

dh

dy

G

KaS

dh

dyy

G

KaS

L

Skyy

L

mG

w

mz

LG

KaSk

Continuous Hydrolysis of Tallow

dh

dy

KaSm

Gy

mx

1

Page 157: Kuliah Diff Eq WS

02

2

00

2

dh

dy

L

KaSm

dh

yd

dh

dy

G

KaS

dh

dyy

G

KaS

L

Skyy

L

mG

w

mz

LG

KaSk

L

mGr

L

Skp

)1( rG

KaSq

w

mzry

r

pqpqy

dh

dyqp

dh

yd 002

2

1)( 2nd O.D.E. with constant coefficients

Complementary function Particular solution

0)(2 pqmqpm Constant at the right hand side

)exp()exp( qhBphAyc Cpqw

mzry

r

pqy p

/

10

0

Continuous Hydrolysis of Tallow

Page 158: Kuliah Diff Eq WS

w

mzry

rqhBphAy 0

01

1)exp()exp(

B.C.

0,

0,0

yHh

xh

dh

dy

q

rymx

1 )exp()exp(

1qhqBphpA

q

rymx

Continuous Hydrolysis of Tallow

0,0 xh 0)(1

CqBpA

q

rBA

0, yHh 0 CBeAe qHpH

KaL

Gk

q

prpqv

1

)()(

)()(qHpHpH

pHpHpH

erCreveA

veCreveB

=C

dh

dyGKaSyKaSmx

Page 159: Kuliah Diff Eq WS

qH

pHqHqh

qH

pHph

er

revee

er

vee

vrw

mzy

)(0

2115.0403.0

1216.0701.0~

403.02

519.0286.0~

069.12

121.1017.1~

0

y

G

L

0,0 yyh

pHqHqhpHphqHpHqH reveeveeerw

mzry

rrevey

)()(

1

1)( 0

0

Continuous Hydrolysis of Tallow

Ka=0.0632 kg glycerine per sec per m3

H=22 my=0

730

66.04

0028.0

32.10

2

S

k

m

qHpH eνr

ve

νr

r

erw

mzy

111

)( pH0

0

L

Skp

L

mGr rortryander� �~~~)1( r

G

KaSq

Page 160: Kuliah Diff Eq WS

Continuous Hydrolysis of Tallow

It shows that the chemical reaction is virtually completed in the bottom 9 m of the column, or 40% of the column.

qH

pHqHqh

qH

pHph

er

revee

er

vee

vrw

mzy

)(0

)exp()exp(1

* qhqBphpAq

rymxy

xyyL

G

w

z

w

z )( 0

0

Page 161: Kuliah Diff Eq WS

)(2

2

xRydx

dyQ

dx

ydP )()( 2211 xycxycyc

Variation of Parameters

)()()()( 2211 xyxuxyxuy p 22221111 ''''' yuyuyuyuy p

0'' 2211 yuyu''' 2211 yuyuy p

'''''''''' 22112211 yuyuyuyuy p

)()()''()''''''''( 2211221122112211 xyuyuRyuyuQyuyuyuyuP

)()'''()'''()''''( 222211112211 xuRyQyPyuRyQyPyyuyuP

0'''

0'''

222

111

RyQyPy

RyQyPy

Page 162: Kuliah Diff Eq WS

Pxyuyu /)('''' 2211

0'' 2211 yuyu

),(

)(]/)([

)'()'(

)()(

)'(/)(

)(0

)('21

2

21

21

2

2

1 yyW

xyPx

xyxy

xyxy

xyPx

xy

xu

),(

)(]/)([

)'()'(

)()(

/)()'(

0)(

)('21

1

21

21

1

1

2 yyW

xyPx

xyxy

xyxy

Pxxy

xy

xu

Variation of Parameters

dxyyW

xyPxxu

),(

)(]/)([)(

21

21

dx

yyW

xyPxxu

),(

)(]/)([)(

21

12

Page 163: Kuliah Diff Eq WS

ExampleSolve )3sec(39

2

2

xydx

yd

092 mm

xBxAyc 3sin3cos

33cos33sin3

3sin3cos),( 21

xx

xxyyW

xxxxy p 3sin3cos3cosln3

1

092

2

ydx

yd

im 3

xxxx����

xBxA��

yyy pc

3sin3cos3cosln3

1

3sin3cos

xxy

xxy

3sin)(

3cos)(

2

1

xdxxx

xu 3cosln3

1

3

3sin3sec3)(1

xdxxx

xu 3

3cos3sec3)(2

Page 164: Kuliah Diff Eq WS

Euler Equation

02

22 By

dx

dyAx

dx

ydx mxxy )( 2

1

)1(')'(

)'(

m

m

xmmxy

mxxy

0)1( mmm BxAmxxmm

0)1(2 BmAm

21, m�mm

21 mm

21

21)( mm xcxcxy

21 mm

1]ln[)( 21mxxccxy

Page 165: Kuliah Diff Eq WS

Our purpose: Use algebraic elimination of the variables until only one differential equation relating two of the variables remains.

Simultaneous Differential EquationsThese are groups of differential equations containing more than one dependent variable but only one independent variable. In these equations, all the derivatives of the different dependent variables are with respect to the one independent variable.

Independent variable or dependent variables?

),(

),(

2

1

yxfdt

dy

yxfdt

dx

elimination of independent variable

),(

),(

2

1

yxf

yxf

dy

dx

Elimination of one or more dependent variablesIt involves with equations of high order and it would be better to make use of matrices.

Page 166: Kuliah Diff Eq WS

Solve

0)23()103(

0)96()6(

22

22

zDDyDD

and

zDDyDD

0)1)(2()5)(2(

0)3()2)(3( 2

zDDyDD

and

zDyDD)5( D

)3( D0)1)(2)(3()5)(2)(3(

0)3)(5()5)(2)(3( 2

zDDDyDDD

and

zDDyDDD

0)23()158()3(

0)1)(2)(3()3)(5(

22

2

zDDDDD

zDDDzDD

0)1311)(3( zDD

Elimination of Dependent Variable

Page 167: Kuliah Diff Eq WS

0)1311)(3( zDD

xxBeAez 311

13

0)96()6( 22 zDDyDD

xAeyDD 11

1322 9

11

136

11

13)6(

= E

x

p EeDD

y 11

13

2 )6(

1

xxc JeHey 32

pxpx epfeDf )()( 11

13p

x

p Eey 11

13

2 )6)1113

()1113

((

1

xx

p AeEey 11

13

11

13

7

4

700

121

Elimination of Dependent Variable

J=2B

xxxHeJeAey 2311

13

7

4

0)23()103( 22 zDDyDDxx

BeAez 311

13

xxxHeJeAey 2311

13

7

4

xxxHeBeAey 2311

13

27

4

Page 168: Kuliah Diff Eq WS

1.25 kg/s of sulphuric acid (heat capacity 1500 J/kg C) is to be cooled in a two-stage counter-current cooler of the following type. Hot acid at 174 C is fed to a tank where it is well stirred in contact with cooling coils. The continuous discharge from this tank at 88 C flows to a second stirred tank and leaves at 45 C. Cooling water at 20 C flows into the coil of the second tank and thence to the coil of the first tank. The water is at 80 C as it leaves the coil of the hot acid tank. To what temperatures would the contents of each tank rise if due to trouble in the supply, the cooling water suddenly stopped for 1h? On restoration of the water supply, water is put on the system at the rate of 1.25 kg/s. Calculate the acid discharge temperature after 1 h. The capacity of each tank is 4500 kg of acid and the overall coefficient of heat transfer in the hot tank is 1150 W/m2 C and inthe colder tank 750 W/m2 C. These constants may be assumed constant.

Example of Simultaneous O.D.E.s

1.25 kg/s

0.96 kg/s

88 C

45 C174 C

20 C40 C80 CTank 1 Tank 2

Page 169: Kuliah Diff Eq WS

When water fails for 1 hour, heat balance for tank 1 and tank 2:

dt

dTVCMCTMCT 1

10

dt

dTVCMCTMCT 2

21

Tank 1

Tank 2

M: mass flow rate of acidC: heat capacity of acidV: mass capacity of tankTi: temperature of tank i

dt

dTTT

dt

dTTT

221

110

B.C.t = 0, T1 = 88 teT 861741

t = 1, T1=142.4 C

dt

dTTe t 2

286174 integral factor, et tetT )12986(1742

t = 1, T2 = 94.9 C

B.C.t = 0, T2 = 45

Example of Simultaneous O.D.E.s

Page 170: Kuliah Diff Eq WS

When water supply restores after 1 hour, heat balance for tank 1 and tank 2:

dt

dTVCMCTtWCMCTtWC ww

22213

Tank 1

Tank 2

W: mass flow rate of waterCw: heat capacity of watert1: temperature of water leaving tank 1t2: temperature of water leaving tank 2t3: temperature of water entering tank 2

dt

dTVCMCTtWCMCTtWC ww

11102

1 2T0 T1 T2

t3t2t1

Heat transfer rate equations for the two tanks:

)ln()ln(

)()()(

2111

21111121 tTtT

tTtTAUttWCw

)ln()ln(

)()()(

3222

32222232 tTtT

tTtTAUttWCw

4 equations have to besolved simultaneously

Example of Simultaneous O.D.E.s

Page 171: Kuliah Diff Eq WS

dt

dTVCMCTtWCMCTtWC ww

22213

dt

dTVCMCTtWCMCTtWC ww

11102

)ln()ln(

)()()(

2111

21111121 tTtT

tTtTAUttWCw

)ln()ln(

)()()(

3222

32222232 tTtT

tTtTAUttWCw

211 )1( ttT

322 )1( ttT

wWC

AU

e11

wWC

AU

e22

Example of Simultaneous O.D.E.s

)1(121 Ttt

)1(232 Ttt

Page 172: Kuliah Diff Eq WS

30975.708.6 22

22

2

Tdt

dT

dt

Td

B.C. t = 0, T1=142.4 C, T2 = 94.9 C

dt

dTCtCTCCTCTtC www

121102 )1(

dt

dTCTCCTCTtCTC www

111032 )1()1()1)(1(

dt

dTCtCTCCTCTtC www

232213 )1(

322

1 )1()( tCTCCCdt

dTCCT www

dt

dTCCC

dt

TdC

dt

dTC ww

222

21 )(

0

23

222

222

22

)1)(1()1(

)1)(1()1(

)22(

TCtCCC

TCCCC

dt

dTCCCCC

dt

TdC

ww

ww

www

t = 1, T2 = 48.8 C

9.3982.126.42 tt BeAeT

Example of Simultaneous O.D.E.s

Eliminating T1

A=0.6; B=54.4