So your integral equation is
$\displaystyle u(t)=1+\int_{0}^{t}s\ln\left(\frac{s}{t}\right)u(s )\,ds,$
and you need to convert to an IVP using the
Leibniz formula, which is, in this case,
$\displaystyle \frac{d}{dt}\int_{a(t)}^{b(t)}f(s,t)\,ds=\frac{db( t)}{dt}\,f(b(t),t)-\frac{da(t)}{dt}\,f(a(t),t)+\int_{a(t)}^{b(t)} \frac{\partial }{\partial t}\,f(s,t)\,ds. $
So, why not just differentiate the original integral equation with respect to t, and use the Leibniz formula on the integral. What do you get?