Does the Euler-Lagrange equation have a series solution?

Question

In classical mechanics the Euler-Lagrange equation of motion is a linear homogeneous ODE of second order, how come we do not have a series solution like other famous differential equations (Legendre, Hermite, Laguerre, Bessel etc etc)?

Answer 1

To add on to Bey's answer, given a functional, the Euler Lagrange equation becomes an ODE which may or may not be linear or homogeneous. It depends entirely on the system you are trying to model. Series solutions may be useful in some cases.