Discussion
Solution of the integral equation
\[ u(t) = a\sin t - 2\int_{0}^{t} \cos(t - s)u(s)\,ds \]
is :
Answer: C
The solution is \(u(t) = ate^{-t}\).
The integral is a convolution, so take Laplace transforms, with \(U = L\{u\}\):
\[ U = \frac{a}{s^{2} + 1} - \frac{2s}{s^{2} + 1}U \]
\[ \begin{aligned} U\frac{s^{2} + 2s + 1}{s^{2} + 1} &= \frac{a}{s^{2} + 1} \\ U &= \frac{a}{(s + 1)^{2}} \end{aligned} \]
Inverting, \(u(t) = ate^{-t}\). Check at \(t = 0\): both sides vanish. The other options do not satisfy the equation: \(a\sin t\) ignores the integral term, and \(at^{2}e^{-t}\) would need \(2a/(s + 1)^{3}\).
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