Solving DE with Homogeneous Sub: v=y/x

[bfusco]

Homework Statement

Consider the DE (x + y)y′ = x − y.
(a) Solve the DE using the homogeneous substitution v = y/x. An implicit solution is acceptable.
(b) We can rearrange the DE into the differential form (y − x) dx + (x + y) dy = 0.
Is this equation exact? If so, find an implicit solution to the equation using our techniques for exact DEs. Show that your solution is equivalent to your answer from part (a). Which method was easier?

The Attempt at a Solution

the question is asking me use the sub. v=y/x which can also be written y=vx and its derivative is y'=v+xv'

-to start distributed that y' to get xy'+yy'=x-y
-then i multiplied 1/x to the whole DE and i get xy'/x + yy'/x = 1-y/x, which reduces to y'+yy'/x=1-y/x, then taking out that y' from the left side of the equation i get, y'(1+y/x)=1-y/x, using the substitutions i get, v+xv'(1+v)=1-v.
-then rewriting v' as dv/dx and bringing that lone v on the left to the right i get, xdv/dx(1+v)=1-2v,
-the dividing by 1+v i get xdv/dx=(1-2v)/(1+v), then i make the equation so that it is in proper form to integrate and i get integral(dx/x)=integral(1+v)/(1-2v)dv

it is at this point I am stuck, I am having difficulty integrating this, and I am not even sure ihave done it right thus far.

 

[bfusco]

-the dividing by 1+v i get xdv/dx=(1-2v)/(1+v), then i make the equation so that it is in proper form to integrate and i get integral(dx/x)=integral(1+v)/(1-2v)dv

it is at this point I am stuck, I am having difficulty integrating this, and I am not even sure ihave done it right thus far.

i would just like to clarify that i know the integral of 1/x, so the issue would be then i guess ln|x|+C=integral(1+v)/(1-2v)dv

 

[haruspex]

integral(1+v)/(1-2v)dv

Haven't checked your working up to that point, but try writing (1+v)/(1-2v) = A + B/(1-2v)