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