1. Class Notes 8: High Order Linear Differential Equation Non Homogeneous
MAE 82 – Engineering Mathematics

2. High Order Differential Equations – Introduction
Solution methods for the particular solution (Nonhomogeneous)
Undetermined Coefficients (polynomials, Exponent, Sin/Cos)
Variation of Parameters (all functions – general method)

3. Method of Undermined Coefficients – Class A

4. Method of Undermined Coefficients – Class B

5. Method of Undermined Coefficients – Class C

6. General Rule for Writing the Correct Form of the Particular Solution

7. General Rule for Writing the Correct Form of the Particular Solution

8. Method of Undermined Coefficients – Example 1

9. Method of Undermined Coefficients – Example 1
Nonhomogeneous term
g(t)
Fundamental solution
Yes/No
class
Particular solution

10. Method of Undermined Coefficients – Example 2

11. Method of Undermined Coefficients – Example 2
Particular solution (Non-homogeneous)
Note for class A

12. Method of Undermined Coefficients – Example 2
Solve for A, B, C

13. Method of Undermined Coefficients – Example 2
Constants:
General Solution of the Differential Equation
Use initial condition
To Solve for C1, C2, C3, C4

14. Method of Variation of Parameters – Review
y1(t), y2(t) are the fundamental solution of
W(s) – Wronskian
Wi(s) - Wronskian where i-th column is replaced by zeros except the last row is 1

15. Method of Variation of Parameters - Example
Given: fundamental solutions
Rewrite the given differential equation in the standard form
Check fundamental solutions

16. Method of Variation of Parameters - Example
Derive the Wronskians

17. Method of Variation of Parameters - Example
General Solution