Hello everyone,

Sorry maybe question quite stupid. I have first order, scalar ODE: $$ \frac{du}{dt}=u_{\infty}(t)-u $$ where $u_{\infty}(t)$ is really heavy, so I'm pretty sure, that there is not any way to resolve classical integral. I need to find value of solution in exttrema $t_X$, specifically in maximum. Obviously that in extremal points: $u_{\infty}(t_X)=u(t_X)$, so if we can get $t_X$, we would easy get $u(t_X)$. We can define initial condition as $u(0) = u_{\infty}(0) = E$, where E - some constant.

Do anyone have any idea how to get $t_X$ without resolving $u(t)$?

Just note that, if we will looking for standard solution: $$u(t) = e^{-t}\left( \int\limits_{t_0}^{t} u_{\infty}(x)e^x dx + C\right) $$ at $t_X$: $$ u_{\infty}(t_X)e^{t_X} = \int\limits_0^{t_X} u_{\infty}(x)e^x dx + C $$ Can we estimate where function $\left(F(t)=u_{\infty}(t)e^{t} \right)$ intersects its integral value and then estimate $C$ from initial condition?

I'll appreciate any help.

Ruben