TITLE: Solve integral (convolution) equation
QUESTION [5 upvotes]: Given a function:
$u(t) = \exp\left( -\frac{At^2}{1+t}\right),$
$A>0, t>0,$
and an equation:
$\frac{d u(t)}{dt} = \int^{t}_0 \phi(t-\tau) u(\tau) d \tau .$
How to find a closed expression for $\phi(t)$?

REPLY [3 votes]: The first terms of the series expansion of the function $\phi(t)$ can be computed as shown in attachment :