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## Homework Statement

Given [itex](\mathcal{L} + k^2)y = \phi(x)[/itex] with homogeneous boundary conditions [itex]y(0) = y(\ell) = 0[/itex] where

\begin{align}

y(x) & = \frac{2}{\ell}\sum_{n = 1}^{\infty} \frac{\sin(k_nx)}{k^2 - k_n^2},\\

\phi(x) & = \frac{2}{\ell}\sum_{n = 1}^{\infty}\sin(k_nx),\\

u_n(x) &= \sqrt{\frac{2}{\ell}}\sum_{n = 1}^{\infty}\frac{\sin(k_nx)}{k^2 - k_n^2},

\end{align}

[itex]\mathcal{L} = \frac{d^2}{dx^2}[/itex], and [itex]k_n = \frac{n\pi}{\ell}[/itex].

If [itex]k = k_m[/itex], there is no solution unless [itex]\phi(x)[/itex] is orthogonal to [itex]u_m(x)[/itex].

## Homework Equations

## The Attempt at a Solution

Why is this?

## Homework Statement

## Homework Equations

## The Attempt at a Solution

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