Find the extremals of the functional $$J(x(t)) = \int\limits_{0}^{2}\frac{[x'(t)]^2}{x^3(t)}dt$$ subject to $x(0) =1$ and $x(2) = 4$. Does the two point boundary problem has a unique solution?

Find all function $x(t)$ which minimizes $$J(x(t)) = \int\limits_{0}^{5}\frac{\sqrt{1+ x'^2}}{\sqrt{x}}dt$$ subject to $x(0) = 0$ and $x(5) = 3$.

P/S: If my computations are true, the Euler - Lagrange equation corresponding to the above functionals are $3x'^2 + 2xx'' -6x' = 0$ and $x'^2 + 2xx''+1 = 0$, respectively. Please help me to find the explicit solutions of the above differential equations, thank you very much.

Your sincerely,