What are methods available for solving 3rd ODE directly without reducing to a 1st order system of IVP?

Question

I have came across the questions: 1, 2, 3.

In these questions you can see different third order ordinary differential equations (ODEs):

$$y'''=f( x, y, y', y''), y(a)=y_0, y'(a)=y_1, y''(a)=y_2.\text{ }(1)$$ As I know a third order equation is rare situation on practice.

I know the Abraham–Lorentz–Dirac–Langevin equation only.

Edit. @Matt gives the link on the KdV equation.

I have found the 3rd ODE which arises in modeling draining or coating fluid flow problems.

Question

What are symbolic and numerical methods available for solving 3rd ODE directly? i.e. without reducing to a 1st order system of inital value problem (IVP)?

Currently, I have found the linearizing tangent transformation which leads the equation (1) into the linear second-order ODE.

Answer 1

A very famous 3rd order equation is the KdV equation.

This equation describes shallow water waves. If you look for traveling wave solutions (keeping the same shape and fixed velocity), then the PDE transforms to a 3rd order ODE.