How to solve this nonlinear ODE either analytically(if solutions exist) or numerically?

Question

Here is the general form of the equation,

$$ a[1+b(y(x)-y_{0})] \frac{d^{2}y}{dx^{2}} + c \frac{dy}{dx} + 1 =0 $$

where $a$, $b$, $c$, $y_{0}$ are constants.

I need a full solution to this ODE, if it exists. If not, how do I go about solving this numerically? Thanks for your help in advance.

Answer 1

The ODE is analytically solvable for $x(y)$, thanks to a special function, the Lambert W function : http://mathworld.wolfram.com/LambertW-Function.html

I doubt that the inverse function $y(x)$ could be expressed on closed form with available standard functions.

Sorry, I have not enough time for a better presentation :