Summary of Separation of Variables — Question 1

Question 1 – Solution

Strategy. Keep one spatial sign convention, then let the original PDE determine whether the remaining equation is first-order decay, oscillation or a spatial boundary problem.

Step 1: Fix the spatial spectrum. The conditions are $X(0)=X(\pi)=0$. At $\lambda=0$, the affine solution must vanish. For $\lambda=-\sigma^2<0$, $X=A\sinh(\sigma x)$ after the left condition, and the right condition forces $A=0$. For $\lambda>0$, the nonzero possibilities are $$\boxed{\lambda_n=n^2,\quad X_n(x)=\sin(nx),\quad n=1,2,\ldots.}$$ All supplied data occupy the $n=2$ mode.

Step 2: Derive the three remaining equations. Substitution gives $a'=-\lambda a/2$ for heat, $b''=-9\lambda b$ for the wave, and $d''=\lambda d$ for Laplace’s equation. At $\lambda=4$, the specified solutions are $$\boxed{H=e^{-2t}\sin(2x),\quad W=\bigl(\cos(6t)+\tfrac 12\sin(6t)\bigr)\sin(2x),\quad P=\frac{\sinh(2y)}{\sinh 2}\sin(2x).}$$

Step 3: Check the original conditions. Each spatial sine vanishes at both ends. At $t=0$, the two time-dependent fields have displacement $\sin(2x)$, while $W_t(x,0)=6(\tfrac 12)\sin(2x)=3\sin(2x)$. The harmonic field is zero at $y=0$ and equals $\sin(2x)$ at $y=1$. Direct second derivatives verify all three PDEs. The heat factor needs one initial value; the wave factor needs position and velocity; the harmonic factor uses two values on different spatial edges.

Step 4: Correct the proposed rule. A positive eigenvalue produces heat decay $e^{-\lambda t/2}$, wave frequency $3\sqrt{\lambda}$, and hyperbolic spatial factors for the harmonic extension. The last factor increases with $y$ because it connects a zero bottom trace to a nonzero top trace; $y$ is not time. The sign of the spatial eigenvalue alone does not identify the remaining factor without the PDE and its data.