Word Count : 632

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Let's solve this step by step using the method of separation of variables for the wave equation. We are given the wave equation: $$ \frac{\partial^2 u(x,t)}{\partial t^2} = 16 \frac{\partial^2 u(x,t)}{\partial x^2}, \quad 0 \le x \le \pi, \ t > 0 $$ with boundary and initial conditions: $$ u(0,t) = u(\pi,t) = 0, \quad u(x,0) = x(\pi-x), \quad u_t(x,0) = 0 $$ ### Step 1: Assume separation of variables Assume: $$ u(x,t) = X(x) T(t) $$ Plug into the PDE: $$ X(x) T''(t) = 16 X''(x) T(t) $$ Divide both sides by $X(x)T(t)$: $$ \frac{T''(t)}{16 T(t)} = \frac{X''(x)}{X(x)} = -\lambda $$ where $\lambda$ is the separation constant. --- ### Step 2: Solve the spatial part $$ X''(x) + \lambda X(x) = 0, \quad X(0)=X(\pi)=0 $$ * The standard solution is: $$ X_n(x) = \sin(n x), \quad n=1,2,3,\dots $$ * And eigenvalues: $$ \lambda_n = n^2 $$ --- ### Step 3: Solve the temporal part $$ T_n''(t) + _______ _________ ____ ______ _________ ______.
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