Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 6 / 2 / a

For a linear spatial operator, let $U(t,s)$ denote the formal homogeneous evolution from time $s$ to time $t$, satisfying ${\partial }_{t}U(t,s)h+D(t)U(t,s)h=0$ and $U(s,s)h=h$. This also allows time-dependent coefficients in the spatial operator. The Duhamel principle statesDifferentiation 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.