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.