= Solution
For a linear spatial operator, let $U(t,s)$ denote the formal homogeneous evolution from time $s$ to time $t$, satisfying $\partial_tU(t,s)h+D(t)U(t,s)h=0$ and $U(s,s)h=h$. This also allows time-dependent coefficients in the spatial operator. The <Duhamel principle> states
$$
\boxed{f(t)=U(t,0)f_0+\int_0^tU(t,s)g(s)\,ds}.
$$
Differentiation of the <integral> contributes $g(t)$ at its upper endpoint and $-D(t)$ times the <integral>, while the initial value is $f_0$. If $D$ is time-independent, one writes $U(t,s)=S(t-s)=e^{-(t-s)D}$. The principle uses <linearity>; an arbitrary nonlinear differential operator would not justify this superposition formula. No analytic construction of $U$ or boundary conditions is required for this formal statement.
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