a calculus of variations question

Question

I want to maximize $\int_0^1(g^2(x)+3g^2(x)g'(x)+2[g'(x)]^2)dx$ subject to g(0)=1, g(1)=0. I can find Euler equation but I can not find g(x) that is maximizes this integral. I have to use calculus of variations. Can you help me?

Answer 1

The Euler Eq. becomes $$2g+6gg'-6gg'-4g''=0 \implies g''(x)-\frac{1}{2} g(x)=0$$ Sp $$g(x)=A \cosh (x/\sqrt{2})+ B\sinh(x/\sqrt{2})$$ $g(0)=1$ gives $A=1$. $g(1)=0$ gives $$A\cosh(1/\sqrt{2})+B \sinh(1/\sqrt{2})=0 \implies B=-\coth(1/\sqrt{2})$$ FGinally we get $$g(x)=\cosh(x/\sqrt{2})-\coth(1/\sqrt{2}) \sinh(x/\sqrt{2})$$