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While working on a physics problem, I have encountered the following DE.

$dx(y-x)+dy(k-y)=0$ where $k$ is a constant.

I have tried various approaches, like trying to concert into a linear DE form, and various other stuff in my knowledge, none if them seem to yield much.

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Write your equation in the form $$\frac{dy}{dx}=\frac{x-y(x)}{k-y(x)}$$ now we substitute $$x=k+t,y=k+v$$ then we get $$\frac{dv(t)}{dt}=\frac{t-v(t)}{v(t)}$$ now we substitute $$v(t)=tu(t)$$ and we get $$t\frac{du(t)}{dt}+u(t)=-\frac{t-tu(t)}{tu(t)}$$ and this is $$\frac{du(t)}{dt}=\frac{-u(t)^2+u(t)-1}{tu(t)}$$ and this can be written as $$\int\frac{\frac{du(t)}{dt}u(t)}{-u(t)^2+u(t)-1}=\int\frac{1}{t}dt$$ Can you finish?

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    $\begingroup$ In your fifth mathematical equation, RHS's numerator should contain $tu(t)$ right ? $\endgroup$ – Esha Manideep Jan 15 '19 at 18:19

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