Not likely. s and t are more likely just throw away variables. (I've seen them typically used in Laplace transform problems.)
The difficulty here is taking the derivative of the integral? Just break it into pieces, as always.
$\displaystyle y = e^{t^2} \int _0^t e^{-s^2}~ds + e^{t^2}$
$\displaystyle y' = \dfrac{d}{dt} (e^{t^2}) \cdot \int _0^t e^{-s^2}~ds + e^{t^2} \cdot \dfrac{d}{dt} \left ( \int _0^t e^{-s^2}~ds \right ) + \dfrac{d}{dt} (e^{t^2} )$
Can you do the derivatives?
-Dan