Laplace II: Steps, Impulses & Convolution · Challenge problem

A drug dose passes through three body compartments in turn (gut, blood, tissue), each emptying into the next at rate $1$: $g'=-g+\delta(t)$, $b'=g-b$, $c'=b-c$, with all three empty before $t=0$. Using Laplace transforms, find the largest amount $c(t)$ that the tissue ever holds.