Strange ODE from my final today?

[Agent M27]

Homework Statement

So I had my final exam today in ODE and I had an equation which appeared to be exact, but was not. I also tried to find a special integrating factor to make it exact, but no success. I then attempted to manipulate it into a linear eq, tried separable variables, even tried to get it into a Bernoulli form, all of which no luck. This was the extra credit question so I know it must be difficult, but I cannot find an example similar to it in our text, "Fundamentals of Differential Equations" by Nagle. Here is the equation and I was told I could use any method I like to solve it.

x(x-y-2)dx+y(y-x+4)=0

Homework Equations

It is not an exact differential, cannot obtain a special integrating factor. Thanks again for any assistance, this problem has been bugging me all day!

The Attempt at a Solution

 

[Mark44]

Homework Statement

So I had my final exam today in ODE and I had an equation which appeared to be exact, but was not. I also tried to find a special integrating factor to make it exact, but no success. I then attempted to manipulate it into a linear eq, tried separable variables, even tried to get it into a Bernoulli form, all of which no luck. This was the extra credit question so I know it must be difficult, but I cannot find an example similar to it in our text, "Fundamentals of Differential Equations" by Nagle. Here is the equation and I was told I could use any method I like to solve it.

x(x-y-2)dx+y(y-x+4)=0

There should be a dy somewhere in this problem.

Homework Equations

It is not an exact differential, cannot obtain a special integrating factor. Thanks again for any assistance, this problem has been bugging me all day!

The Attempt at a Solution

 

[Agent M27]

Crap you're correct Mark, the equation actually read as x(x-y-2)dx+y(y-x+4)dy=0. I have determined that I need to do some sort of substitution, using v=y/x, which ought to transform it into some sort of separable variables. However, I am still running into trouble attempting to separate into dv=dx. Here is where I get stuck:

\frac{dy}{dx}=-\frac{x(x-y-2)}{y(y-x+4)}

dividing by x

=-\frac{(1-y/x-2/x)}{y/x(y/x-1+4/x)}=-\frac{(x-y-2)}{(y^{2}/x-y+4y/x)}

substituting v=y/x, y=vx, \frac{dy}{dx}=\frac{dv}{dx}x+v and some rearranging and I get stuck...

\Rightarrow \frac{dv}{dx} = -\frac{(x-vx-2)}{(v^{2}x-vx^{2}+4vx)}-\frac{v}{x}

Do I now need to come up with another substitution? Thanks for you help.

Joe