- #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.

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- Thread starter Math10
- Start date

- #1

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

None

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.

- #2

Homework Helper

2022 Award

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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.

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

Science Advisor

Homework Helper

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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

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Thank you so much for the help, Hallsoflvy.

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