tags: Method Of Variation Of Parameter Differential Equations Complementary Functions Particular Integral 3rd Semester Engineering Mathematics BTCEH CSE

1. NAME:SAMPADKAR
STUDENTCODE:
BWU/BTA/22/225
GROUP:D
SECTION:D2
COURSE:DIFFERENTIAL
EQUATIONANDCOMPLEX
ANALYSIS
COURSECODE:BSCM301
SESSION:2023-24

2. Parameters in
Differential
Equations

3. Contents

4. 1.Introduction
2.Background
3.Methodology
4.Example
5.Advantages
and limitations

5. Introduction

6. Definition
Variation of parameters is a method used to find particular solutions to non-
homogeneous linear differential equations. It is a general method for finding
a particular solution of a differential equation by replacing the constants in
the solution of a related equation by functions and determining these
functions so that the original differential equation will be satisfied.
When is it used?
In cases where the method of undetermined coefficients fails, particularly
when the non-homogeneous term is a combination of functions.

7. Background

8. Linear Differential Equations
A linear equation or polynomial, with one or more terms, consisting of the derivatives of the
dependent variable with respect to one or more independent variables is known as a linear
differential equation.
A general first-order differential equation is given by the expression:
𝑑𝑦
𝑑𝑥
+ 𝑃𝑦 = 𝑄 where y is a function and dy/dx is a derivative.
The solution of the linear differential equation produces the value of variable y.
Non-Homogeneous Equations
A nonhomogeneous linear differential equation is an ordinary differential equation (ODE) in which
the dependent variable and its derivatives are present on one side of the equation, and there is a
non-zero function on the other side. The general form of a nonhomogeneous linear differential
equation of the second order is:
y′′(x)+p(x)y′(x)+q(x)y(x)=r(x) , where p and q are co-efficients
We can find both the Complementary Function (CF) and the Particular Integral (PI) of the equation.

9. Methodology

10. Steps in Variation of Parameters
1.Find the Complementary Solution (Homogeneous
Part): Solve the associated homogeneous differential
equation.
2.Find the Particular Solution (Particular Part): Assume
a particular solution and determine its undetermined
coefficients.
3.Combine Solutions: Form the general solution by
combining the complementary and particular solutions.

11. Example Problem

12. Find a general solution to the following differential equation.
𝑦′′ − 2𝑦′ + 𝑦=
𝑒𝑡
𝑡2+1
𝑚2 − 2𝑚 + 1 = 0
𝑌𝑐(𝑡) = (𝑐1 + 𝑐2𝑡)𝑒𝑡

13. Now, we have to replace the c1 and c2 with u(t) and v(t) so that the equation becomes:
𝑌𝑝 = (𝑢 𝑡 + 𝑣 𝑡 𝑡)𝑒𝑡
Step 2: Finding The Wronskian
𝑊 = 𝑒𝑡
𝑡𝑒𝑡
𝑒𝑡
𝑒𝑡
+ 𝑡𝑒𝑡
= 𝑒𝑡(𝑒𝑡 + 𝑡𝑒𝑡) - 𝑒𝑡(𝑡𝑒𝑡)
= 𝑒2𝑡
Step 3: Finding PI
u(t) = −
t𝑒𝑡𝑒𝑡
𝑒2𝑡 𝑡2+1
dt = −
1
2
log(𝑡2
+ 1)

14. v(t) =
𝑒𝑡
𝑒𝑡
𝑒2𝑡 𝑡2 + 1
dt = 𝑡𝑎𝑛−1(𝑡)
Therefore, The PI Of This Equation,
𝑌𝑝 = −
1
2
𝑒𝑡
log(𝑡2
+ 1) +𝑡𝑒𝑡
𝑡𝑎𝑛−1
(𝑡)
So, The General Solution Of This Equation,
𝑌(𝑡) = (𝑐1 + 𝑐2𝑡)𝑒𝑡
−
1
2
𝑒𝑡
log(𝑡2
+ 1) +𝑡𝑒𝑡
𝑡𝑎𝑛−1
(𝑡)

15. Advantages and
Limitations

16. Pros and Cons of Variation of
Parameters
•Advantages: Flexibility in handling various
non-homogeneous terms.
•Limitations: Complexity may increase
with the nature of the differential equation.

17. Thank you