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Dustinsfl
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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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