Solve $ h(t) = \int_{0}^{t} e^{ - (\eta + \mu) s } \{\mu + \eta \, \, h(t-s)^2 \} ds. $

Question

Suppose that $h(t) : [0, \infty) \rightarrow [0, \infty)$ satisfies, $$ h(t) = \int_{0}^{t} e^{ - (\eta + \mu) s } \{\mu + \eta \, \, h(t-s)^2 \} ds, $$ where $\eta, \mu \geq 0$. I need to prove that, if $\mu \neq \eta$, then, $$ h(t) = \frac{ \eta e^{\eta t} - \eta e^{ \mu t } }{ \eta e^{\eta t} - \mu e^{\mu t} }. $$ Would you have some hint?

Answer 1

This is hint in the form of a suggestion, as one way to approach the problem. Since you don't have to derive the expression for $h(t)$, you could first check that it satisfies the integral equation, and second show that the integral equation has a unique solution. One way to show uniqueness would be to derive from the integral equation, an ordinary differential equation satisfied by $h$, since we have a standard uniqueness theorem for ode's.