= Solution
Write $T_s$ for the fixed surface <temperature> and $\theta=\overline T-T_s$. The <heat capacity> per unit volume is $\rho_0c_p=k/\kappa$. Per unit horizontal area the layer stores <sensible heat> $(k/\kappa)d\theta$. Its increase equals the input less the output:
$$
\boxed{\frac{kd}{\kappa}\frac{d\overline T}{dt}=F-E(t).}
$$
The same equation holds for $\theta$ because $T_s$ is constant. It assumes the supplied constant-property, fixed-thickness layer, without internal <heat> production, <latent heat>, or mechanical contributions to its <heat> budget. The infinite horizontal area is handled by taking the budget per unit area.
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