Is there any general method to solve a Second order ODE with variable coefficient of the form $a(x)y''(x)+b(x)y'(x)+c(x)y(x)=0$ Is there any general method to solve a Second order ODE with variable coefficient of the form $a(x)y''(x)+b(x)y'(x)+c(x)y(x)=0$
 A: The equation you have is a second order linear homogeneous ODE. 
It can be rewritten as 
$$ y''(x) + b_1(x)\, y'(x) + c_1(x) y(x) = 0, $$
where 
$b_1 = b/a$ and $c_1 = c/a$ with $a \neq 0$.
A general solution of second order linear homogeneous ODE can be obtained in multiple ways, including integration and series expansion. 
I propose to read this article to get an overview of possible ways to solve it.
A: In order to successfully approach the problem submitted, the 'Reduction of Order' technique should prove quite useful.  Here is a short video that will demonstrate how to solve such a problem with this method:  https://www.youtube.com/watch?v=oQSFW8BIrY0&t=627s
A: The answer is NO.
Even an equation as simple as Airy's equation:
$$
y''-xy=0,
$$
can NOT be solved. Meaning, its solutions can not be expressed in closed form, as expressions of simple functions, their compositions and integrals.
The only method which always produces an answer is the method of power series, which works if all the coefficients are expressible as power series (i.e., real analytic).
