Hey! I had the same problem as you and I asked a question about the same topic yesterday ! 

You should take a look here: https://www.mapleprimes.com/questions/229311-Dsolve-Return-Either-Nothing-Or-Random

It has Euler-Lagrange with two independent variables.

With this simple problem, I've been able to solve a more complex system by following the same steps.

Hope this help !