= Solution
\b[Steady state.] In the <heat equation>, time independence makes the second spatial <derivative> zero. The two endpoint <temperatures> therefore determine the linear profile
$$
\boxed{T_\infty(x)=T_0x/l.}
$$
Insulation is along the sides; the endpoint reservoirs can still exchange <heat> with the rod.
\b[Transient solution.] Subtract the steady profile before applying <separation of variables>: let $U=T-T_0x/l$. Then $U_t=\nu U_{xx}$, $U(0,t)=U(l,t)=0$ and $U(x,0)=-T_0x/l$ for $0<x<l$. Writing $U=X(x)\Theta(t)$ gives
$$
\frac{\Theta'}{\nu\Theta}=\frac{X''}{X}=-\mu,\qquad X''+\mu X=0,\qquad\Theta'+\nu\mu\Theta=0.
$$
The homogeneous <Dirichlet boundary conditions> exclude $\mu\leq0$ and select the <eigenfunctions> $\sin(n\pi x/l)$ with <eigenvalues> $\mu_n=(n\pi/l)^2$, $n\geq1$. Their <orthogonality> gives the <Fourier sine series> coefficients
$$
b_n=\frac2l\int_0^l\left(-\frac{T_0x}l\right)\sin\frac{n\pi x}l\,dx=\frac{2T_0(-1)^n}{n\pi}.
$$
Here <integration by parts> uses $\int_0^l x\sin(n\pi x/l)\,dx=-l^2(-1)^n/(n\pi)$. Thus
$$
\boxed{T(x,t)=\frac{T_0x}l+\frac{2T_0}\pi\sum_{n=1}^\infty\frac{(-1)^n}{n}\sin\frac{n\pi x}l\,e^{-\nu n^2\pi^2t/l^2},\qquad t>0.}
$$
Every summand satisfies the <heat equation> and the homogeneous endpoint conditions, while the steady profile supplies the prescribed endpoint values. The initial <Fourier sine series> equals $-T_0x/l$ in the open interval. At $(l,0)$ the initially cold rod and the suddenly imposed hot endpoint are incompatible; the formula describes the prescribed data for $t>0$ and the initial limit at interior points.
\b[Heat absorption.] Integrating in $x$ kills the even modes. With $a=\nu\pi^2/l^2$, the total <heat> relative to the initial state is
$$
H(t)=\frac{\gamma T_0l}2-\frac{4\gamma T_0l}{\pi^2}\sum_{j=0}^\infty\frac{e^{-a(2j+1)^2t}}{(2j+1)^2}.
$$
Consequently the net rate of <heat> absorption is
$$
\boxed{\frac{dH}{dt}=\frac{4\gamma\nu T_0}l\sum_{j=0}^\infty e^{-\nu\pi^2(2j+1)^2t/l^2}.}
$$
For each $t\geq\varepsilon>0$, any fixed number of differentiated modes is bounded by a polynomial in $n$ times $e^{-a\varepsilon n^2}$. These bounds have finite sums, proving <absolute convergence>, <uniform convergence> on such time intervals and the validity of the integration and differentiation. In particular the displayed rate is finite for every $t>0$, positive when $T_0>0$, and tends to zero as equilibrium is approached. The same answer follows from the <heat equation> balance $H'=\gamma\nu[T_x(l,t)-T_x(0,t)]$; this subtracts loss at the cold endpoint from input at the hot endpoint.
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