This is with regards to the first section of Wikipedia article Duhamel's principle (revision from July 2012). I want to see if I am understanding this.
Basically the inhomogeneous equation says that heat is being added at a rate of $f(x,t)$, so at each point $x$, $f(x,t)\,dt$ of heat is being added to the system.
And you can think of this as a set of systems $u_t(x,t)-\Delta u(x,t)=0$, $u(x,t_0)=f(x,t_0)$ for all $t_0\in(0,\infty)$, which says the initial heat distribution is $f(x,t)$, and no heat is being added, and if we integrate these solutions over $t_0$ we get the inhomogeneous solution...
It's the third paragraph I am a bit uncomfortable with. Can anyone offer some help?