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?



Is there a closed-form solution for $\frac{dy}{dx} = 1 + \frac{a}{y} + \frac{b}{x}$?
One thing you can try is to select numerical values for $a$ and $b$ (say $a=b=1$ or $a=b^{-1}=2$) and then find a power series solution $y=1+\sum\_{k>0}c\_k(x-1)^k$ and inspect the coefficients $c\_k$. With $a=b=1$ they are
$$ 3, 2, 4, 11, 35, \frac{721}{6}, \frac{18163}{42}, \frac{540391}{336}, 
 \frac{98091}{16}, \frac{26684211}{1120},\dotsc $$
If there was a straightforward solution in terms of standard functions then you would expect that the coefficients would be arithmetically simple, with no large primes involved in $c\_k$ when $k$ is small. (Large primes can appear in the numerator by accident, but not so easily in the denominator.) Also, the terms $k!c\_k$ seem to be integers, so we can look them up in the Online Encyclopedia of Integer Sequences. In fact, the prime $286927$ appears in the numerator of $c\_{10}$ and OEIS does not recognise the sequence $k!c\_k$ so that is not promising for the existence of an elementary solution.


