Module – 2

Differential Equation

2.1 First order differential equations

An exact differential equation is formed by differentiating its primitive solution without any other process mdx + Ndy =0 is said to be an exact differential equation if it is satisfies the following conditions

Method for solving exact differential equation

Step I :-Integrate m with respect to x keeping by constant

Step II :-Integrate with respect to y only those terms of N which do not contain x.

Step III :-Result of step I + Result of step II= constant

Solve.

Ans.Here m=

N=

Since,    the given equation is exact.

Now, (Term of N is not containing x)dy=c

2.2 Exact differential equation

An exact differential equation is formed by differentiating its primitive solution without any other process mdx + Ndy =0 is said to be an exact differential equation if it is satisfies the following conditions

Method for solving exact differential equation

Step I :-Integrate m with respect to x keeping by constant

Step II :-Integrate with respect to y only those terms of N which do not contain x.

Step III :-Result of step I + Result of step II= constant

Solve.

Ans.Here m=

N=

Since,    the given equation is exact.

Now, (Term of N is not containing x)dy=c

2.3 Linear differential equation

A differential equation of the form is called linear differential equation where p and q are function of x(but not of y) or constant

Note:-  is called integrating factor

Working rule

Step 1:- Convert the given equation to the standard form of linear differential equation i.e.

Step 2:- Find the integrating factor

Step 3:- then the solution is

Example:- Solve (x+1)

Ans.

IF=

Then solution is

2.4 Bernoulli’s equation application to Electrical circuits

The equation of the form where p and q are constant or function of x can be reduced to the linear form on dividing by and substituting

Example. Solve

Ans. Given

Dividing by xy we get

Equation (1) becomes

Where,

Its solution is

References

1. Ordinary and Partial Differential equations by J. Sihna Ray and S Padhy, Kalyani Publishers

2. Advance Engineering Mathematics by P.V.O’NEIL, CENGAGE

3. Ordinary Differential Equation by P C Biswal , PHI secondedition.

4. Engineering Mathematics by P. S. Das & C. Vijayakumari, Pearson. N.B:Thecourseisof3creditwith4contacthours.