Solving an Equation: a"+bxy'+cy*x^2=0

[ziad1985]

I have this equation , I feel like the solution should be rather easy to have, but somehow it evades me, maybe I'm mistaken, what type is this equation?
any explicit solutions?
ay"+bxy'+cy*x^2=0
a,b,c are just constant.

 

[Astronuc]

That's a basic Second-Order Ordinary Differential Equation
http://mathworld.wolfram.com/Second-OrderOrdinaryDifferentialEquation.html

y"(x) + p(x) y'(x) + q(x) y(x) = 0, where p(x) = ax, q(x) = bx²,

and find one solution then use the Wronskian to find the other, IIRC.

It's been a long time since I've done it.

 

[ziad1985]

As simple as that?
For a second I thought It could only be solved by a Power Series or something like that, I think I'm becoming rusty!

 

[HallsofIvy]

As simple as that?
For a second I thought It could only be solved by a Power Series or something like that, I think I'm becoming rusty!

Depends on what you mean by simple. That is, of course, a "Second-Order Ordinary Differential Equation" with variable coefficients. Power series is, in fact, the most common method of solving such an equation.