Exact solution to $\frac{\mathrm dy}{\mathrm dx} = 1 + \frac{a}{y} + \frac{b}{x}$

Question

This equation arises from my attempt to study the quasi-steady state of a cross-diffusive system. $$\frac{\mathrm{d}y}{\mathrm{d}x} = 1 + \frac{a}{y} + \frac{b}{x},$$ where $2>a > 1> b > 0$, $x,y \in \mathbb{R}^{+}$.

I learned that it is a special case of the Chini equation, which typically does not have a closed form solution. But it looks so simply to not have a closed form solution! On the other hand, I have tried many different methods and substitution to try to obtain its closed form in vain.

Could someone please help suggest a way to do this?

Answer 1

Hint:

$\dfrac{dy}{dx}=1+\dfrac{a}{y}+\dfrac{b}{x}$

$y\dfrac{dy}{dx}=\left(1+\dfrac{b}{x}\right)y+a$

This belongs to an Abel equation of the second kind.