$$ I(y,t) = \int_{-a}^tF(x,y,t)dx $$$$ F(x,y,t) = \frac{1}{f_1(t)}\exp\left\{ -f_2^2(y,t-x)-x^2- \frac{f_3^2(y,t-x)}{f_1^2(t-x)} \right\} $$$$ f_1^2(t) = b-Q^2(t) $$$$ f_3(y,t) = c-(y-Q(t))Q(t) $$$$ f_2(y,t) = y-Q(t) $$$$ Q(t) = exp(-t) \, or \, exp(-t^2) $$
For different values of the parameter t, the functional I(y,t) gives Gaussian-like functions depending on y. I am wondering how the maximum of this function will behave depending on the parameter t. To do this, I need to find the derivative $\frac{\partial I}{\partial y}=0$. I know how to solve the problem numerically, but I would like to get an analytical expression. The derivative with respect to t can be found using the Leibniz integral rule, but it is of little use, so my problem is to express $y_{max}(t) \leftarrow \frac{\partial I}{\partial y}=0$.