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[Differential Equation] One-Dimentional Spring Problem

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For $x \in (0, L), t > 0$ ,

$\begin{align*} u_{tt} &= u_{xx} \\ f(x) &= u(x, 0) \\ u_t(x, 0) &= 0 \\ u(0, t) &= u(L, t) = 0 \\ \end{align*}$

Suppose form of solution

$u_n(x, t) = \sum_{n=1}^{\infty} c_n u_n$

$u_n(x, t) = X_n(x)T_n(t)$

From $u_{tt} = u_{xx}$

$\dfrac{X''(x)}{X(x)} = \dfrac{T''(t)}{T(t)} = - \lambda$

or

$\begin{align*} X''(x) + \lambda X'(x) &= 0 \\ T''(t) + \lambda T'(t) &= 0 \\ \end{align*}$

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