Find all solutions?

1. Oct 8, 2014

Math10

1. The problem statement, all variables and given/known data
Find all solutions of xy'=x^3+(1-2x^2)y+xy^2.

2. Relevant equations
None

3. The attempt at a solution
Here's my work:

xy'=x^3+y-2x^2*y+xy^2
xy'=x(x^2-2xy+y^2)+y
xy'=x(x-y)^2+y
y'=(x-y)^2+y/x

2. Oct 8, 2014

pasmith

You can see from the last line that $y(x) = x$ is one solution, although there may be others. But your rearrangement is not separable, so you are unlikely to make further progress.

The left hand side of the original is $xy'$. There's a $y$ on the right, so bringing that across makes the LHS $xy' - y = x^2(y/x)'$, so the substitution $v = y/x$ is worth considering.

3. Oct 9, 2014

HallsofIvy

Staff Emeritus
Good idea!

If v= y/x, then y= xv so that y'= xv'+ v. xy'=x^3+y-2x^2*y+xy^2 becomes x^2v'+ xv= x^3+ xv- 2x^3v+ x^3v^2.

4. Oct 9, 2014

Math10

Thank you so much for the help, Hallsoflvy.