Solving non-linear first order differential equation

Question

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.

Answer 1

$$\frac{dy}{dx}= \frac{\frac{-a~y}{x}}{1- ~b~\left(\frac{1}{x}\right)^n\left(\frac{y}{1-y}\right)^m}$$ Consider $x'$ instead of $y'$. Then this is Bernoulli's differential equation: $$-ay\frac{dx}{dy}= {x- ~b~x^{1-n}\left(\frac{y}{1-y}\right)^m}$$

Answer 2

$$\frac{dy}{dx}= \frac{\frac{-a~y}{x}}{1- ~b~\left(\frac{1}{x}\right)^n\left(\frac{y}{1-y}\right)^m}$$

$$\frac{dx}{dy}= -\frac{x}{a~y}+\frac{b}{a} \frac{y^{m-1}}{(1-y)^m} x^{1-n}$$

$$x^{n-1}\frac{dx}{dy}= -\frac{x^n}{a~y}+\frac{b}{a} \frac{y^{m-1}}{(1-y)^m}$$ 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.