Solving a differential equation

[Lengalicious]

Homework Statement

Solve
$$(1+bx)y''(x)-ay(x)=0$$

Homework Equations

The Attempt at a Solution

I'm used to solving homogeneous linear ODE's where you form a characteristic equation and solve that way, here there is the function of x (1+bx) so how does that change things?

 

[frzncactus]

Would dividing both sides by 1+bx help?

 

[Lengalicious]

Ok so if I did that then what? I can define a characteristic equation such that

$$r^2-\frac{a}{1+bx}=0$$

and $$r=\pm\sqrt{\frac{a}{1+bx}}$$

where $$b^2-4ac = 4a(1+bx) > 0$$

so a solution is $$y=ce^{rx}$$ but that doesn't satisfy the ODE so its not correct?

 

[LCKurtz]

You can't use constant coefficient methods on a DE like this with variable coefficients. Perhaps there is a clever substitution that will help, or maybe not. Problems like this are typically solved with series solutions, especially if you know $a$ and $b$. Where did this equation come from? If it's from a text, the recent material may give a hint how to solve it.