x dy + y dx = x^m*y^n dx , m ? n - 1

[ttn181]

How do I find the general solution to this differential equation?

I just don't know where to start. :/

 

[galactus]

It's been a while. Hope it is not too late. I just noticed your problem.

Anyway, this looks like a Bernoulli Equation if we rearrange:

\(\displaystyle xdy=x^{m}y^{n}dx-ydx\)

\(\displaystyle xdy=(x^{m}y^{n}-y)dx\)

Separate variables:

\(\displaystyle \frac{dy}{dx}=\frac{x^{m}y^{n}-y}{x}\)

\(\displaystyle \frac{dy}{dx}+\frac{y}{x}=x^{m-1}y^{n}\)

The general solution would then be:

\(\displaystyle y=\frac{1}{\left(\frac{x^{m}(n-1)}{n-m-1}+C_{1}x^{n-1}\right)^{\frac{1}{n-1}}}\)