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**Lancet** I have a homogeneous ODE which I'm trying to solve:

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

Book answer:

$\displaystyle c(x + y)^2 = xe^{\frac{y}{x}}$

Here's what I'm doing:

$\displaystyle v + x \frac{dv}{dx} = \frac{v^2 + 1}{v - 1}$

$\displaystyle x \frac{dv}{dx} = \frac{v + 1}{v - 1}$

$\displaystyle \int \frac{1}{x} dx = \int \frac{v - 1}{v + 1} dv$

$\displaystyle ln(x) = \int 1 - \frac{2}{v + 1} dv$

$\displaystyle ln(x) = v - 2ln(v + 1) + c$

$\displaystyle x = \frac{e^v}{(v + 1)^2} c$

$\displaystyle x = \frac{e^{\frac{y}{x}}}{(\frac{y}{x} + 1)^2} c$

$\displaystyle cx = \frac{e^{\frac{y}{x}}}{(\frac{y + x}{x})^2}$

...and this obviously looks nothing like the solution.

Where did I make a mistake?