# Extremals of functionals

1. 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?

2. 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.

BTW, you should preferably write $J(x)$, as $J$ is a function of $x$. Writing $J(x(t))$ is misleading or meaningless, as $t$ is a dummy integration variable. –  Pietro Majer Feb 23 '13 at 14:04
I will not cover the details of the calculation because this sounds a lot like a homework question but to cover the broader details, to find the extremals of the functional $J(y),$ they must satisfy the Euler-Lagrange equation $$\frac{d}{dt}\frac{\partial L}{\partial x'}=\frac{\partial L}{\partial x}$$ where $L$ is the integrand of the functional $J$. For example, for $1.$, we have $$\frac{d}{dt}\frac{\partial L}{\partial x'}=\frac{\partial L}{\partial x}\Rightarrow \frac{2x''}{x^3}=-\frac{3(x')^2}{x^2}$$ and for the second, $$\frac{(x'')}{(1+(x')^2)^{1/2}\sqrt{x}}=\frac{\sqrt{1+(x')^2}}{2x^{3/2}}.$$ With uniqueness, the addition of an arbitrary constant does not give uniqueness.