I have an equation like this:
$$te^{t} = \int\nolimits_0^t e^\tau u(\tau)d\tau$$
I don't really know how to solve it.. Would it be possible to differentiate both sides of the equation? If so, how can I do it - is it like differentiate one side and then the other or is it more complex? And if it is possible what conditions need to be satisfied in order to do it?
If we differentiate the left hand side we get $e^t+t e^t$, if we differentiate the right hand side we get $e^t u(t)$ by the fundamental theorem of calculus. Then we see
$$u(t)=t+1$$
An we are done. I think we need to demand that $u(t)$ is continuous to get uniqueness. In general $u(t)=t+1$ almost everywhere.