Find the function $y(x)$ that gives a stationary value of the functional

$$I[y]=\int_{0}^{1}\left(y^{\prime 2}+y y^{\prime}+y^{\prime}+y^{2}+y x^{2}\right) d x$$

subject to the boundary conditions $y(0)=-1$ and $y(1)=e-e^{-1}-\frac{3}{2}$.