Basic concepts of first order differential equations

8/12/2019 Basic concepts of first order differential equations

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Differential Equations

(Mathematics-3)

Prepared By-

Rootvesh S MehtaSapan Shah

26/5/2010,General,Maths-3 1


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Unit-1

Basics of Differential Equations

Defini t ion-

Differential Equation --A differential equation is anequation containing an unknown function

and its derivatives. Examples are

n unknown function

and its derivatives.

26/5/2010,General,Maths-3 2

43

3

6 3d y dy

ydx dx

2

2 0

y y

x x


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Differential equations can be classified in two parts

(1) Ordinary differential equations

(2) Partial differential equations

An Ordinary differential equation is an equation which

involves ordinary derivatives .

Is an example of O.D.E

A partial differential equation is an equation which

involves partial derivatives . Is an

example of P.D.E

25/5/2010,Maths-3,General 3

2

2 0y yx x

43

3 6 3

d y dy

ydx dx


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Order of Differential Equation

25/5/2010,Maths-3,General

The order of the differential equation is order of the highest

derivative in the differential equation.

Differential Equation ORDER

32 xdxdy

0932

2

ydx

dy

dx

yd

36

4

3

3

y

dx

dy

dx

yd

1

2

3

4


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Degree of Differential Equation

Differential Equation Degree

032

2

aydx

dy

dx

yd

36

4

3

3

y

dx

dy

dx

yd

03

53

2

2

dx

dy

dx

yd

1

1

3

The degree of a differential equation is power of the highest

order derivative term in the differential equation.


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Definition--The solution of a differential equation is a

function which satisfies given equation

types of solutions of differential equations

1) General solution2) Particular Solution

, 6

Solution of a differential equation

25/5/2010,Maths-3,General


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General Solution- The solution of differential equation

is called general if the no. of arbitrary constants equals to

the order of differential equationParticular solution- If we assign particular value to arbitrary

constant in general solution then it is called particular

solution

Example - y=3x+c is solution of the 1st order differential

equation ,here its a general solution .

Now ,if we take c=5 in y=3x+c then its a particular

solution.


3dx

dy


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Formation of Differential equation


We can form differential equation using following steps

I. Construct function from given data

II. Differentiate as many times as the no. of arbitrary

constants given and eliminate that all arbitrary constants.

Example-Find the differential equation from y=ax-a2

Solution Here we have y=ax-a2___{1}. Here we have only one

arbitrary constant a .Now, Differentiating w.r.to x we get dy/dx =a _____{2}

From Eq.---1 and Eq.2 we get


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Which is the required differential

equation


,0

2

y

dx

dyx

dx

dy

Exercise- Construct a differential equation ofthe circle having radius a and centre

origin.


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Solution of first order and first degree differential equation

DefinitionA differential equation having first order and

first degree is called first order and first degree

differential equation

The general form of first order and first degree is

M(x,y)dx + N(x,y)dy

Ordy/dx = f(x,y) .



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(A)Separable variable differential equations

(B) Equations reducible to separable variable

(C)Homogenous differential equations(D)Exact Differential equations

(E) Non Exact differential equations

(F) Linear differential equations

(G)Bernoulli Differential equations


We will solve following types of O.D.E of first order and first

Degree


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Thank You

Basic concepts of first order differential equations

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