First Order Ordinary Linear Differential Equations

• Ordinary Differential equations does not include partial derivatives.

• A linear first order equation is an equation that can be expressed in the form

Where p and q are functions of x

Types Of Linear DE:

1. Separable Variable2. Homogeneous Equation3. Exact Equation4. Linear Equation

The first-order differential equation

,dy f x ydx

is called separable provided that f(x,y) can be written as the product of a function of x and a function of y.

(1)

Separable Variable

Suppose we can write the above equation as

)()( yhxgdxdy

We then say we have “ separated ” the variables. By taking h(y) to the LHS, the equation becomes

1 ( )( )

dy g x dxh y

Integrating, we get the solution as

1 ( )( )

dy g x dx ch y

where c is an arbitrary constant.

Example 1. Consider the DE ydxdy

Separating the variables, we get

dxdyy

1

Integrating we get the solution askxy ||ln

or ccey x , an arbitrary constant.

Homogeneous equationsDefinition A function f(x, y) is said to be homogeneous of degree n in x, y if

),(),( yxfttytxf n for all t, x, yExamples 22 2),( yxyxyxf is homogeneous of degree

)sin(),(x

xyxyyxf

is homogeneous of degree

2.

0.

A first order DE 0),(),( dyyxNdxyxM

is called homogeneous if ),(,),( yxNyxMare homogeneous functions of x and y of the same degree.

This DE can be written in the form

),( yxfdxdy

The substitution y = vx converts the given equation into “variables separable” form and hence can be solved. (Note that v is also a new variable)

Working Rule to solve a HDE:1. Put the given equation in the form

.Let .3 zxy

)1(0),(),( dyyxNdxyxM

2. Check M and N are Homogeneous function of the same degree.

5. Put this value of dy/dx into (1) and solve the equation for z by separating the variables.

6. Replace z by y/x and simplify.

dxdzxz

dxdy

4. Differentiate y = z x to get

EXACT DIFFERENTIAL EQUATIONS

A first order DE 0),(),( dyyxNdxyxMis called an exact DE if

M Ny x

The solution id given by:

cNdyMdx(Terms free from ‘x’)

Solution is:

cyxy

cdyy

ydx

ln2

2

cNdyMdx

Integrating FactorsDefinition: If on multiplying by (x, y), the DE

0M dx N dy

becomes an exact DE, we say that (x, y) is an Integrating Factor of the above DE

2 2

1 1 1, ,xy x y

are all integrating factors of

the non-exact DE 0y dx x dy

( )

M Ny x g x

N

( )g x dxe

is an integrating factor of the given DE

0M dx N dy

Rule 1: I.F is a function of ‘x’ alone.

Rule 2: I.F is a function of ‘y’ alone.

If , a function of y alone,

then( )h y dy

e is an integrating factor of the given DE.

xN

yM

M yh

is an integrating factor of the given DE

0M dx N dy

Rule 3: Given DE is homogeneous.

NyMx1

Rule 4: Equation is of the form of

021 xdyxyfydxxyf

Then,

NyMx

1

Linear EquationsA linear first order equation is an equation that can be expressed in the form

1 0( ) ( ) ( ), (1)dya x a x y b xdx

where a1(x), a0(x), and b(x) depend only on the independent variable x, not on y.

We assume that the function a1(x), a0(x), and b(x) are continuous on an interval and that a1(x) 0on that interval. Then, on dividing by a1(x), we can rewrite equation (1) in the standard form

( ) ( ) (2)dy P x y Q xdx

where P(x), Q(x) are continuous functions on the interval.

Rules to solve a Linear DE:1. Write the equation in the standard form

( ) ( )dy P x y Q xdx

2. Calculate the IF (x) by the formula

( ) exp ( )x P x dx 3. Multiply the equation by (x).

4. Integrate the last equation.

Applications

)mTα(TdtdT

αdt)T(T

dT

m

•Cooling/Warming lawWe have seen in Section 1.4 that the mathematical formulation of

Newton’s empirical law of cooling of an object in given by the linear first-order differential equation

This is a separable differential equation. We have

or ln|T-Tm |=t+c1

or T(t) = Tm+c2et

In Series Circuits

i(t), is the solution of the differential equation.

)(tERidtdi

dtdqi

)t(EqC1

dtdqR

Since

It can be written as