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veneficus5
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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