- #1

Dr.Doom

- 8

- 0

## Homework Statement

Find the general solution of the differential equation:

(x+y)y'=x-y

## Homework Equations

I want to solve this as a homogeneous differential equation, so our equations are:

v=[itex]\frac{y}{x}[/itex], y=vx, [itex]\frac{dy}{dx}[/itex]=v+x[itex]\frac{dv}{dx}[/itex]

## The Attempt at a Solution

I need to get this into the form [itex]\frac{dy}{dx}[/itex]=F([itex]\frac{y}{x}[/itex]), so I rewrite it as y'=[itex]\frac{x-y}{x+y}[/itex]. Dividing by x I get, y'=[itex]\frac{1-\frac{y}{x}}{1+\frac{y}{x}}[/itex]. From here, I substitute to get v+x[itex]\frac{dv}{dx}[/itex]=[itex]\frac{1-v}{1+v}[/itex]. When looking at the solution manual, however, it says that it should be in the form x(v+1)v'=-(v

^{2}+2v-1) before I integrate. I don't see how i can get it into this form. Also, it gives the answer as y

^{2}+2xy-x

^{2}=C. I can correctly solve this differential equation using different methods, but I would really like to know how to solve this using the homogeneous method. I would really appreciate any help with this. Thanks!