Solving non-linear first order differential equation I have the following differential equation problem but I couldn't proceed any further -
$$\frac{dy}{dx}= \frac{\frac{-a~y}{x}}{1- ~b~\left(\frac{1}{x}\right)^n\left(\frac{y}{1-y}\right)^m}$$
where, $x \in [0,1] ~\text{and} ~ y \in [0,1]$
But I can't solve it down.
I have tried $y= uy_1 ~~~
\text{where}, ~ y_1 = x^{\frac{n}{m}}~$, but it didn't help.
Wolfram gives the solution as -
$$y(x) = c_1 \exp( \int  \frac{a}{x - b \frac{x^{m - n + 1}}{(-x + 1)^m}} \, dx) $$
How to simplify the integral?
I just wanted a hint that whether it can be solved? If yes, please just tell me what am I doing wrong.
 A: $$\frac{dy}{dx}= \frac{\frac{-a~y}{x}}{1- ~b~\left(\frac{1}{x}\right)^n\left(\frac{y}{1-y}\right)^m}$$
Consider $x'$ instead of $y'$. Then this is Bernoulli's differential equation:
$$-ay\frac{dx}{dy}= {x- ~b~x^{1-n}\left(\frac{y}{1-y}\right)^m}$$
A: $$\frac{dy}{dx}= \frac{\frac{-a~y}{x}}{1- ~b~\left(\frac{1}{x}\right)^n\left(\frac{y}{1-y}\right)^m}$$
$$\frac{dx}{dy}= -\frac{x}{a~y}+\frac{b}{a} \frac{y^{m-1}}{(1-y)^m} x^{1-n}$$
$$x^{n-1}\frac{dx}{dy}= -\frac{x^n}{a~y}+\frac{b}{a} \frac{y^{m-1}}{(1-y)^m}$$
Let $X=x^n$
$$\frac{1}{n}\frac{dX}{dy}= -\frac{X}{a~y}+\frac{b}{a} \frac{y^{m-1}}{(1-y)^m}$$
This is a first order lineaer ODE. Solving it involves an integral which closed form is an hypergeometric function.
$$X(y)=C\:y^{-n/a}+\frac{b~n}{a~m+n}\:y^m\:{_2F_1}(m\:,\:m+\frac{n}{a}\:;\:m+\frac{n}{a}+1\:;\:y)$$
$C$ is an arbitrary constant.
$_2F_1$ denotes the Gauss hypergeometric function.
$$x(y)=\left(C\:y^{-n/a}+\frac{b~n}{a~m+n}\:y^m\:{_2F_1}(m\:,\:m+\frac{n}{a}\:;\:m+\frac{n}{a}+1\:;\:y) \right)^{1/n}$$
$y(x)$ is the inverse function of the above.
