# Forced Vibrating Membranes and Resonance - Separation of Variables

## Homework Statement

In the midst of Forced Vibrating Membranes and Resonance Utt = c^2*delsquared(U) + Q(heat source)

Arrive at eigenfunction series solution where the coefficients are given by
d^2/dt^2 (A_n) + c^2*lambda_n*A_n = q_n

## Homework Equations

according to the book, I am supposed to arrive here

A_n = c1 * cos (c*sqrt(lambda_n)*t) + c2 * sin (c*sqrt(lambda_n)*t) <--- homogeneous part of solution + particular solution ---> integral from 0 to t of (q_n * sin (c*sqrt(lambda_n)*(t-tau)) / (c*sqrt(lambda_n)) with respect to tau.

## The Attempt at a Solution

Now when I try the variation of parameters (given my two homogeneous solutions already which match with the book),
I get
-cos(c*sqrt(lambda_n)*t) * integral of ( q_n * sin (c*sqrt(lambda_n)*t) / c*sqrt(lambda_n) + -sin(c*sqrt(lambda_n)*t) * integral of ( q_n * sin (c*sqrt(lambda_n)*t) / c*sqrt(lambda_n)

How do I reconcile this into the single integral term, the definite integral that I am supposed to get?

Thanks

fzero
Homework Helper
Gold Member
-cos(c*sqrt(lambda_n)*t) * integral of ( q_n * sin (c*sqrt(lambda_n)*t) / c*sqrt(lambda_n) + -sin(c*sqrt(lambda_n)*t) * integral of ( q_n * sin (c*sqrt(lambda_n)*t) / c*sqrt(lambda_n)

How do I reconcile this into the single integral term, the definite integral that I am supposed to get?

Thanks

If you write these integrals in terms of the dummy variable $$\tau$$, you can simplify the expression using the trig identities for $$\sin(A\pm B)$$. You should double check your signs, since your expression doesn't seem to reduce to the answer that you claim.