TITLE: Is there a closed-form solution for $\frac{dy}{dx} = 1 + \frac{a}{y} + \frac{b}{x}$?
QUESTION [8 upvotes]: I am looking for an exact solution for the following special case of Chini Equation with $2\geq a > 1 > b > 0, x, y \in \mathbb{R}^+$,
$$\frac{dy}{dx} = 1 + \frac{a}{y} + \frac{b}{x}$$
I have tried to approach this using multiple methods and substitutions, but none has gotten me far. I know that Chini equation tends not to have closed form solution, but this looks so simple to not have a closed form solution!
Could someone please suggest a way to try obtaining the solution?

REPLY [9 votes]: Maple does not find a closed-form solution.  Note that the substitution $y = 1/u$ produces the Abel DE $$u' = - a u^3 - \frac{b+x}{x} u^2$$ but Maple can't find a closed-form solution for that either: it doesn't seem to fall into one of the known exactly solvable classes.
EDIT: You might also note that scaling the dependent and independent variables by $y = cY$, $x = cX$ gives you a similar equation for $Y(X)$ with $a$ and $b$ replaced by $a/c$ and $b/c$ respectively.  So effectively there is just one parameter here.