How to Use the Method of Undetermined Coefficients for Differential Equations?

[Tom McCurdy]

Homework Statement

Set up but do not solve for the appropriate particular solution yp for the differential equation

y''+25y=–1xcos(5x)

using the Method of Undetermined Coefficients (primes indicate derivatives with respect to x).

In your answer, give undetermined coefficients as A, B, etc.

Homework Equations

Possible Derivatives

• sin(x)
• cos(x)
• xsin(x)
• xcos(x)

The Attempt at a Solution

This seems like the solution should simply be

$$A*x*sin(5*x)+B*x*cos(5*x)+C*cos(5*x)+D*sin(5*x)$$

 

[HallsofIvy]

What solution are you talking about? The problem said "Set up but do not solve for the appropriate particular solution yp for the differential equation
y''+25y=–1xcos(5x)
using the Method of Undetermined Coefficients"

Normally, to get a solution of the form "x cos(5x)" you would use (Ax+B) cos(5x)+ (Cx+ D) sin(5x). However, in this case cos(5x) and sin(5x) are solutions to the homogeneous equation. Multiply what you would normally use by x: (Ax²+ Bx)cos(5x)+ (Cx²+ Dx)sin(5x).