# General solution methods for third-order linear differential equation with variable coefficients

I'm trying to solve the following IVP:

$$y''' + ay''+by'+cy=0$$,

with initial conditions $$y(x_{0})=y_{0},y'(x_{0})=y_{1},y''(x_{0})=y_{2}$$,

where $$y,a,b$$, and $$c$$ are all functions of the independent variable $$x$$.

I have seen some methods for solving second degree linear ODEs with variable

coefficients, but most methods get complicated really quick for the third order case.

Can anyone recommend a method and/or give me a reference for trying to solve this

type of equations?

Edit: Although I could try to solve this equation using numerical methods, a close-form

solution would be more valuable to me. The way I am thinking about this problem is

to get a particular solution $$y_{p}$$ and then replace $$y=y_{p}z$$. I understand

something like this would allow me to reduce the order of the ODE, and I could use

the methods for a 2d order ODE with variable coefficients from there. The problem

then is how to come up with the particular solution $$y_{p}$$.

Thanks!

• What sort of solution do you want? Closed form? Does not exist in most cases. Series solution? Easy, with recursive formulas for the coefficients, provided $a,b,c$ are analytic at $x_0$. Numerical solution? Such as Picard's method... – GEdgar May 23 at 17:49
• I was hoping to find a close form solution. However, a Numerical solution could also be useful. – Nick May 23 at 18:01
• It depends on how the $a$, $b$ and $c$ are. As @GEdgar said, a closed form solution most likely does not exist. A numerical solution, however, is easy to obtain. There are a lot of numerical techniques you could code. Are you familiar with MATLAB or Mathematica? – Thales May 23 at 18:04
• Yes, I am familiar with numerical methods to solve ODEs. But now that I look at my problem in more detail, I would rather see if there is a close form solution. I will modify my question to give more details. – Nick May 23 at 19:47