Forced Vibrating Membranes and Resonance - Separation of Variables

[veneficus5]

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]

-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.