# How do you solve differential equations in the form of $ay'' + by' +cy = d$?

I've only recently started learning how to work with differential equations. I'm about to start on linear 2nd order DE, but I wondered - what approach is taken to solve equations of the form:

$ay'' + by' +cy = d$

where $a$,$b$,$c$,$d$ and $y$ are all functions of $x$?

I'm sure I'll get to that point soon, but I just thought I'd throw out an enquiry. It was a very useful question to ask when I started learning about first orders linears.

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Out of many ways, the series solution is a useful one. – Tapu Sep 20 '12 at 0:56
eqworld.ipmnet.ru/en/methods/methods-ode/Khorasani2003.pdf claims that can solve general 2nd order linear ODE analytically. Does it true? – doraemonpaul Sep 20 '12 at 23:23

In general, these equations do not have closed-form solutions, or even an integral formula for solutions. In the homogeneous case ($d=0$) where $a$, $b$, $c$ are rational functions there is an algorithm due to Kovacic to decide whether there are "Liouvillian" solutions (roughly speaking, those are solutions that can be expressed in terms of the exponential function, algebraic functions and integrals). For example, $y'' + (x^3+1) y = 0$ has no Liouvillian solutions (and, as far as I know, no closed-form solutions of any kind). However, this is not something for beginners to deal with.