PRESENTATION IS ABOUT SOLUTION OF DIFFERENTIAL EQUATION BY FOLLOWING THREE METHODS 1. VARIATION OF PARAMETER 2. CAUCHY S EQUATION 3. UNDETERMINED COEFFICIENT AND BASIC FORMULAS AND SOLVED EXAMPLES ARE INCLUDED

1. SARDAR VALLABHBHAI PATEL
INSTITUTE OF
TECHNOLOGY
PREPARED BY : PANCHAL PARTH HITESHKUMAR (18MEMECG003)
GUIDED BY : H. R. PATEL

2. TOPICS :
1. METHOD OF VARIATION OF PARAMETERS
2. CAUCHY’S LINEAR EQUATION
3. METHOD OF UNDETERMINED COEFFICIENTS

3. METHOD OF VARIATION OF PARAMETERS :
This method can be used for finding the particular integral yp
yp =y1 ʃ R(x) dx + y2 ʃ R(x) dx + y3 ʃ R(x) dx + .......
Where y1, y2, y3,… are basis of the solution.
For y”  m2  two roots  y1, y2
W = |y1 y2 | ,
|y1’ y2’|
W1 = |0 y2 |
|1 y2’|
W2 = |y1 0 |
|y1’ 1 |

4. For y”’  m3  three roots  y1, y2, y3
W = |y1 y2 y3 | ,
|y1’ y2’ y3’|
|y1” y2” y3”|
W1 = |0 y2 y3 |
|0 y2’ y3’|
|1 y2” y3”|
W2 = |y1 0 y3 |
|y1’ 0 y3’|
|y1” 1 y3”|
W3= |y1 y2 0|
|y1’ y2’ 0|
|y1” y2” 1|

5. EXAMPLE-1:

6. EXAMPLE-2:

8. CAUCHY’S LINEAR EQUATION :
The ODE of the form ,
is called Cauchy Linear equation.
To convert the about equation into equation with constant coefficient, take
Where θ = d/ dz

9. EXAMPLE-3:

12. General solution is

13. METHOD OF UNDETERMINED COEFFICIENTS :
This method can be used to find particular integral only if linearly independent
derivatives of Q(x) are finite in number.
This restriction implies that Q(x) can only have the terms such as k, xn, eax, sin ax, cos
ax and combination of such terms where k and a are constant and n is a positive
integer.
However, when Q(x) = 1/x or tan x or sec x, etc. , this method fails, since each function
has an infinite number of linearly independent derivatives.

14. Some of the choices of the particular integrals are given below :
In the table A0, A1, A2,……..,An are coefficients to be determined. To obtain the
values of these coefficients, we use the fact that the particular integral satisfies the
given differential equation.

15. EXAMPLE-4:

17. EXAMPLE-5: