= Solution
{c}
Write $T=T_s+T_u$. The transient satisfies the <heat equation> and homogeneous mixed <boundary conditions>
$$
(T_u)_t=k(T_u)_{xx},\qquad
(T_u)_x(0,t)=0,\qquad T_u(2\ell,t)=0,\qquad
T_u(x,0)=-\frac Qk(2\ell-x).
$$
The left condition is a <Neumann boundary condition> and the right one a <Dirichlet boundary condition>. <Separation of variables>, $T_u=X(x)G(t)$, gives
$$
\frac{G'}{kG}=\frac{X''}{X}=-\mu^2,\qquad
X''+\mu^2X=0,\qquad G=e^{-k\mu^2t}.
$$
The condition $X'(0)=0$ removes the <sine> term. The condition $X(2\ell)=0$ then selects
$$
\mu_n=\frac{(2n+1)\pi}{4\ell},\qquad
X_n(x)=\cos(\mu_nx),\qquad n=0,1,2,\ldots.
$$
There is no nonzero mode with $\mu=0$, because a linear $X$ satisfying both conditions is zero. There is no growing mode either: <integration by parts> gives $\int_0^{2\ell}(X')^2dx=\mu^2\int_0^{2\ell}X^2dx$, so every nonzero mode has $\mu^2>0$.
These <Sturm-Liouville eigenfunctions> have <orthogonality> and squared <norm>
$$
\int_0^{2\ell}\cos(\mu_nx)\cos(\mu_jx)\,dx=\ell\,\delta_{nj}.
$$
Expand the initial transient in this half-integer <Fourier cosine series>. Its coefficients are
$$
\begin{aligned}
A_n&=-\frac Q{k\ell}\int_0^{2\ell}(2\ell-x)\cos(\mu_nx)\,dx\\
&=-\frac Q{k\ell}\frac{1-\cos(2\ell\mu_n)}{\mu_n^2}
=-\frac{16Q\ell}{k\pi^2(2n+1)^2}.
\end{aligned}
$$
The integration formula follows by <integration by parts>; the cosine at the endpoint is zero. Thus the <half-integer cosine expansion of a linear profile> gives
$$
\boxed{T(x,t)=\frac Qk(2\ell-x)
-\frac{16Q\ell}{k\pi^2}\sum_{n=0}^\infty
\frac{\cos\!\left(\frac{(2n+1)\pi x}{4\ell}\right)}{(2n+1)^2}
\exp\!\left(-\frac{(2n+1)^2\pi^2kt}{16\ell^2}\right).}
$$
At $t=0$ the <Fourier series> reproduces $-T_s$ everywhere, including the endpoints; its $n^{-2}$ coefficient decay gives <uniform convergence>. For every $t>0$ the exponential factors allow termwise differentiation and verify the <heat equation> and both <boundary conditions>. The sudden switch-on of nonzero <heat flux> is incompatible with the spatial derivative of the constant initial temperature at the corner $(0,0)$. Accordingly, the positive-time boundary derivative must not be inferred by differentiating the initial <Fourier series> term by term at that corner.
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