Find the extremal of $J[y]=\int_1^2(y'^2+2yy'+y^2) dx $

Question

I have to find the extremal for the following functional: $$J[y]=\int_1^2(y'^2+2yy'+y^2) dx $$ such that $y(1)=1$ and $y(2)$ is arbitrary.

I got it to be equal $e^{x+1}$. Is that correct?

Answer 1

Notice that $$J[y,y'] = \int_1^2 (y' + y)^2 dx.$$ Clearly this is always non-negative; it can only reach zero if $y' + y = 0$ which gives $y = e^{-x+1}$ (after taking into acct the boundary condition) so it appears you're missing a sign, though that may have just been a typo.