Find the general solution in implicit form $dy/dx =xy/(x-y)$

Question

I am trying to find the solution to the following differential equation in implicit form, and I seem not to be getting anywhere:

$$\frac{dy}{dx} = \frac{xy}{x-y}$$

This is not separable, but I tried separating them anyway, such that on the $dx$ side there were only $x$ terms, and then I figured the $x$ terms of the $dy$ side I could hold constant since $x$ is not a function of $y$, but I realized I don't actually know what $y$ is so I can't say that for sure.

Any ideas?

EDIT: I've tried it on Wolfram and it won't/can't do it.

EDIT: The actual problem I had to solve was:

$$\frac{dy}{dx} = \frac{xy}{x^2-2y^2}$$

I thought this was of the same form as the equation I wrote above, and that If I knew how to solve that, I could solve this. I see however that they are actually quite different. Thank you for the responses.

Answer 1

$$\frac{dy}{dx} = \frac{xy}{x-y}$$

Looks like Abel's equation

$$g'=-\frac {g^2} y +g^3$$

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For this equation $y=tx$ is simply enough for solving it $$y' = \frac{xy}{x^2-y^2}$$ $$t'x+t=\frac {t}{1-t^2}$$ $$t'x=\frac {t^3}{1-t^2}$$ And that equation is seperable $$\int\frac{1-t^2} {t^3}dt=\ln|x|+K$$ $$\frac{1}{2t^2} +\ln|t|+\ln|x|=K$$ $$\boxed{\frac{x^2}{2y^2} +\ln|y|=K}$$

Answer 2

Hint

In a comment, you wrote that the problem is $$y' = \frac{xy}{x^2-y^2}$$ which is a totally different story.

Write the equation as $$x'= \frac{x^2-y^2}{xy}$$ and let $x=y z$ which makes $$z+y z'= \frac{z^2-1}z\implies y z'=-\frac 1 z$$ which is now very simple to solve.

Answer 3

If the equation is : $$\frac{dy}{dx} = \frac{xy}{x^2-y^2}$$ this is a first order non-linear ODE of homogeneous kind. Thus, easy to solve with the usual change $y(x)=xu(x)$.

If the equation is : $$\frac{dy}{dx} = \frac{xy}{x-y}$$ it's something else entirely.

Let $x=\frac{1}{u(y)}\quad ;\quad dx=-\frac{du}{u^2}$ $$\frac{dx}{dy} = \frac{1}{y}-\frac{1}{x} = -\frac{1}{u^2}\frac{du}{dy}=\frac{1}{y}-u$$ $$\frac{du}{dy}=\left(u(y)\right)^3-\frac{1}{y}\left(u(y)\right)^2$$ This is an Abel's ODE of the first kind. These kind of equations are generally not solvable in terms of a finite number of standard functions : https://arxiv.org/ftp/arxiv/papers/1503/1503.05929.pdf