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- #1

\[

y'' + k^2y = \phi(x)

\]

with homogeneous boundary conditions \(y(\ell) = 0\) and \(y(0) = 0\) by expanding \(y(x)\) and \(\phi(x)\)

\begin{align*}

y(x) &= \sum_na_nu_n(x)\\

\phi(x) &= \sum_nb_nu_n(x)

\end{align*}

in the eigenfunctions of \(L = \frac{d^2}{dx^2}\) where \(Lu_n(x) = -k^2u_n(x)\) and \(u_n\) satisfies the homogeneous boundary conditions.

How am I supposed to use the definitions of \(y(x)\) and \(\phi(x)\) to solve this problem?