Separation of Variables — Question 7

Question 7 – Solution

Strategy. A change of spatial coordinate can simplify separation, but the original equation determines the correct integration weight.

Step 1: Transform the spatial equation carefully. Substitution gives $XT'/x=(xX')'T$. Nonzero anchors yield $(xX')'=-\lambda X/x$ and $T'=-\lambda T$. Since $X'=Z_s/x$, one has $xX'=Z_s$ and $(xX')'=Z_{ss}/x$. Multiplying the spatial equation by $x$ gives $$\boxed{Z_{ss}+\lambda Z=0,\qquad Z(0)=Z(1)=0.}$$ The physical endpoints $1,e$ become $0,1$, not an interval of length $e-1$.

Step 2: Recover the modes in the original variable. The transformed Dirichlet problem has $\lambda_n=n^2\pi^2$ and $Z_n(s)=\sin(n\pi s)$ for $n\ge 1$. Thus $$\boxed{u_n(x,t)=C\sin(n\pi\log x)e^{-n^2\pi^2t}.}$$ Indeed $xX_n'=n\pi\cos(n\pi\log x)$ and $(xX_n')'=-n^2\pi^2X_n/x$. Multiplication by the time factor gives exactly $u_t/x$. At $x=1,e$ the sine values vanish. Zero and negative eigenvalues produce only the trivial Dirichlet factor.

Step 3: Derive the weighted energy quotient. Multiplication of $-(xX')'=\lambda X/x$ by $X$ and integration give $$\boxed{\lambda\int_1^e\frac{X^2}{x}\,dx=\int_1^e x(X')^2\,dx.}$$ The endpoint term vanishes. For a nonzero factor the denominator is positive; zero numerator would force a constant and then the zero field. Hence $\lambda>0$. The weight $1/x$ comes from the coefficient of $u_t$; dropping it would change the identity and the associated eigenvalue quotient.

Step 4: Verify orthogonality, norms and nodes. For distinct modes $m,n$, subtracting the two integrated spatial identities cancels the derivative terms and yields $(\lambda_n-\lambda_m)\int_1^e X_nX_m\,dx/x=0$. Thus their weighted inner product vanishes. With $s=\log x$, $$\int_1^e\frac{X_n^2}{x}\,dx=\int_0^1\sin^2(n\pi s)\,ds=\frac 12.$$ For $n=3$, the interior zeros occur at $s=1/3,2/3$, hence at $x=e^{1/3},e^{2/3}$. Equal spacing in $s$ is not equal spacing in $x$.