Coupled ODEs from Euler Lagrange eq

[Dustinsfl]

Given \(F = A(x)u_1^{'2} + B(x)u'_1u'_2 + C(x)u_2^{'2}\).
\[
\frac{\partial F}{\partial u_i} - \frac{d}{dx}\left[\frac{\partial F}{\partial u_i'}\right] = 0
\]
From the E-L equations, I found
\begin{align*}
\frac{d}{dx}\left[2Au_1' + Bu_2'\right] &= 0\\
\frac{d}{dx}\left[2Cu_2' + Bu_1'\right] &= 0\\
2Au_1' + Bu_2' &= D\\
2Cu_2' + Bu_1' &= E
\end{align*}

Just a general question. Is this DE going to be extremely difficult to solve or is it relativily trivial?

 

[Ackbach]

It's going to be relatively trivial. Let $v_{1}=u_{1}'$ and $v_{2}=u_{2}'$. Then it's not actually a DE in the $v_{i}$'s, but just an algebraic equation. This happens because there are no $u_{i}$ terms, but only their derivatives. So you can solve for the $v_{i}$'s using your favorite method, and then at least symbolically integrate the result.