- #1

2h2o

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## Homework Statement

Find a general solution

## Homework Equations

[tex] (x+y)\frac{dy}{dx} = x-y [/tex]

## The Attempt at a Solution

[tex]

\frac{dy}{dx} = \frac{x-y}{x+y}

[/tex]

let v=y/x

y=xv

[tex]

\frac{dy}{dx} = v+x\frac{dv}{dx}

[/tex]

now,

[tex]

v+x\frac{dv}{dx} = \frac{x-xv}{x+xv} [/tex]

[tex]= \frac{1-v}{1+v} [/tex]

[tex]= \frac{1}{1+v} - \frac{v}{1+v} [/tex]

[tex]=\frac{1}{1+\frac{y}{x}}-\frac{\frac{y}{x}}{1+\frac{y}{x}}[/tex]

Which takes me back to where I started if I clear the denominators, so I'm spinning the wheels. This looks very familiar to me, but I don't recall what it is or what to do with it. Separate the variables, then integrate? Could do it, wrt v, by parts, but isn't there a more efficient way?

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