I have been working on solving Euler-Lagrange Equation problems in differential equations, specifically in Calculus of Variations, but this one example has me stuck. I am probably making mistakes in my integration.
I am supposed to solve the Euler-Lagrange equation given that $f(t,x(t),x^{\prime}(t))=f(t,u,v)=\frac{\sqrt{1+v^2}}{u}$.
I know that $f_u=-\frac{\sqrt{1+v^2}}{u^2}$ and $f_v=\frac{v\sqrt{1+v^2}}{u(1+v^2)}$. Plugging this into the Euler-Lagrange Equation, $f_u-\frac{d}{dt}f_v$ gives $$\frac{\sqrt{1+(x^{\prime})^2}}{x^2}-\left(\frac{x^{\prime}\sqrt{1+(x^{\prime})^2}}{x(1+(x^{\prime})^2)}\right)^{\prime}=0.$$
I cannot figure out from here how to solve it. Have I made a mistake up to now? Could someone help me out in actually solving this for x?