find the extremal for
\int^{T}_{0} (\dot{x}^2 + 2x\dot{x} + 2x^2) dt
when x(0) = 1 and T = 2 i.e. x(2) free.

i get x = Ae^{\sqrt{2}t} + Be^{-\sqrt{2}t}
and the first condition x(0) = 1 gives A + B = 1
then when i apply the transversality condition T = 2
ie. \frac{df}{d\dot{x}}(2) = 0
it gets really complicated. can someone please tell me if i have the correct equation for x?