Find all solutions?

  • Thread starter Math10
  • Start date
  • #1
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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.
 

Answers and Replies

  • #2
pasmith
Homework Helper
2,058
684

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 [itex]y(x) = x[/itex] 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 [itex]xy'[/itex]. There's a [itex]y[/itex] on the right, so bringing that across makes the LHS [itex]xy' - y = x^2(y/x)'[/itex], so the substitution [itex]v = y/x[/itex] is worth considering.
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
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966
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
301
0
Thank you so much for the help, Hallsoflvy.
 

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