= Solution
For the spatial mode with <eigenvalue> $\kappa_n=(n\pi/L)^2$, the temporal <ordinary differential equation> is
$$
\boxed{T_n'+D\left(\frac{n\pi}{L}\right)^2T_n=0.}
$$
Its <general solution> is
$$
\boxed{T_n(t)=C_n\exp\left(-\frac{Dn^2\pi^2}{L^2}t\right),\qquad C_n\in\mathbb R.}
$$
The printed reference to “each solution $X(x)$ ... in part (b)” refers to (b)(i). The zero spatial mode gives $u=0$ irrespective of $T$ and contributes no additional nonzero solution.<Discrete mathematics, continuous mathematics, and probability examination (2015) by University of Oxford>[Printed page 7, Question 6(b)(ii), internal part reference]{ref}
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