Forced Vibrating Membranes and Resonance - Separation of Variables

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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
 

Answers and Replies

  • #2
fzero
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-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 [tex]\tau[/tex], you can simplify the expression using the trig identities for [tex]\sin(A\pm B)[/tex]. You should double check your signs, since your expression doesn't seem to reduce to the answer that you claim.
 

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