Solve xy'=x^3+(1-2x^2)y+xy^2: Solutions & Tips

[Math10]

Homework Statement

Find all solutions of xy'=x^3+(1-2x^2)y+xy^2.

Homework Equations

None

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
Now I'm stucked. Please help me.

 

[pasmith]

Homework Statement

Find all solutions of xy'=x^3+(1-2x^2)y+xy^2.

Homework Equations

None

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
Now I'm stucked. Please help me.

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.

 

[HallsofIvy]

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.

 

[Math10]

Thank you so much for the help, Hallsoflvy.