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

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)

Method of Undermined Coefficients – Class A

Method of Undermined Coefficients – Class B

Method of Undermined Coefficients – Class C

General Rule for Writing the Correct Form of the Particular Solution

General Rule for Writing the Correct Form of the Particular Solution

Method of Undermined Coefficients – Example 1

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

Method of Undermined Coefficients – Example 2

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

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

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

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

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

Method of Variation of Parameters - Example Derive the Wronskians

Method of Variation of Parameters - Example General Solution