Separation of Variables — Question 1

Question 1 – Solution

Strategy. Evaluate at a nonzero anchor value of one factor before deriving equations valid even at zeros of the other.

Step 1: Derive a global spatial identity. Substitution yields $XT'=\kappa X''T+rXT$. Choose $t_0\in J$ with $T(t_0)\ne 0$. Evaluating there gives, for every $x\in I$, $$\kappa X''(x)=\left(\frac{T'(t_0)}{T(t_0)}-r\right)X(x).$$ Set $\lambda=(r-T'(t_0)/T(t_0))/\kappa$. Then $\boxed{X''+\lambda X=0}$ on all of $I$, including any zeros of $X$.

Step 2: Recover the time equation and the converse. Substituting $X''=-\lambda X$ into the product identity gives $X[T'-(r-\kappa\lambda)T]=0$. Choose $x_0$ with $X(x_0)\ne 0$ to obtain $$\boxed{T'=(r-\kappa\lambda)T\quad\text{on }J.}$$ The constant is independent of both variables; a different anchor must give the same value because $X$ is not identically zero. Conversely, these two ODEs imply the original product identity directly, without any divisions. Thus every such pair of factors produces a PDE solution.

Step 3: Treat zeros and the excluded trivial field. The time equation gives $T(t)=C e^{(r-\kappa\lambda)(t-t_0)}$. For a nontrivial factor $C\ne 0$, it never vanishes. Spatial zeros are allowed; the identities above remain valid there. A factor identically zero produces $u\equiv 0$, which solves the PDE but determines no separation constant from its chosen factors. The trivial field must be retained separately, not mistakenly assigned restrictions obtained by division.

Step 4: Verify a concrete mode and its normalization. For $X=\sin(2x)$, $X''=-4X$, so $\lambda=4$ and $$u(x,t)=C\sin(2x)e^{(r-4\kappa)t}.$$ Its nonzero amplitude grows if $r>4\kappa$, is constant if $r=4\kappa$, and decays if $r<4\kappa$. Direct differentiation verifies the PDE, including at $x=n\pi/2$. Replacing $(X,T)$ by $(aX,T/a)$ for $a\ne 0$ leaves $u$ unchanged; the two factors are not individually unique until a normalization is chosen.