Nonlinear differential equation issue

[Fooze]

Homework Statement

This is just a nonlinear differential equation. All I have to do it solve it, though it is an initial value problem as well.

2*y*y' + y^{2} = t

Initial value:

y(0) = -1.

The Attempt at a Solution

This should be easy, but it doesn't seem easily separable to me. If you try separating it, then you end up with a integral of y^{2}dt, which doesn't seem right to me at all.

Maybe I'm missing something very fundamental?

And how do you get the Latex stuff to show up right? (sorry I'm new to trying to do Latex on a forum)

Thanks for any assistance. I know what I'm doing, I think, once I get started...

 

[tiny-tim]

Welcome to PF!

Hi Fooze! Welcome to PF!

(try using the X² tag just above the Reply box )

2*y*y' + y^{2} = t

Hint: do you notice something about the lhs?

try a substitution ​

 

[Fooze]

I haven't tried it yet on paper (don't have any handy for another 45 minutes or so) but should I be thinking like a u(t) = y(t)²?

That will leave me with a u' + u = t, I think? Then it's linear and really easy with an integrating factor. Is that the right thought?

 

[tiny-tim]

I haven't tried it yet on paper (don't have any handy for another 45 minutes or so) but should I be thinking like a u(t) = y(t)²?

That will leave me with a u' + u = t, I think? Then it's linear and really easy with an integrating factor. Is that the right thought?

Yup!

erm … can't you solve that in your head? ​

 

[Fooze]

I don't trust myself at all solving integrating factor problems in my head. I had issues in the beginning of the semester with them... and it's just recently that I've been able to do them properly at all. ;)

But thanks so much for your help... I should be able to get it from there!