Finding extremals over y in two ways

Question

We can find the extremal of $$\int_0^1(\frac{1}{2}y'^2+yy'+y'+y)dx$$ amongst all y with $y(0)=1$ by imposing the natural boundary condition $\frac{\partial F}{\partial x}=y'+y+1=0$ at $x=1$.Solving this gives $$y=2e^{-x}-1$$
How do we do this by imposing the boundary conditions $y(0)=1, y(1)=\frac{3}{2}+a$ and minimising over a rather than the natural boundary condition?