Variable Separation: Solving x^2y'=1-x^2+y^2-x^2y^2

[erok81]

Homework Statement

$$x^2y'=1-x^2+y^2-x^2y^2$$

Homework Equations

n/a

The Attempt at a Solution

I am trying to separate the y terms on one side and the x terms on the other so I can solve this differential equation. I've tried everything I can think of, but cannot get them on their respective sides.

Any hints starting in the right direction?

 

[annoymage]

Homework Statement

$$x^2y'=1-x^2+y^2-x^2y^2$$

Homework Equations

n/a

The Attempt at a Solution

I am trying to separate the y terms on one side and the x terms on the other so I can solve this differential equation. I've tried everything I can think of, but cannot get them on their respective sides.

Any hints starting in the right direction?

$$x^2y'=(1-x^2)+(1-x^2)y^2$$

 

[erok81]

That was actually what I tried in the beginning.

Next I tried dividing everything by x^2 to get y' alone. Then tried subtracting (1-x^2)y^2. Then spent the next ten minutes moving things back and forth until I started over.

This is really stupid, but say I am dividing both sides by y^2. It has to go into all three pieces, correct? It doesn't just get canceled out one the one side, does it?

 

[annoymage]

hmm, you can factorize $$(1-x^2)$$

$$(1-x^2)(1+y^2)$$

;P

 

[erok81]

Haha, thanks.

I love spending forever on some problems, only to find out the easiest method is the correct answer. I always get stuck on the easiest ones. I don't get it. $$(1-x^2)(1+y^2) = 1-x^2+y^2-x^2y^2$$ aka the original problem. I don't think they make factoring problems easier than that.