# Nonlinear ODE

• Jan 19th 2011, 07:15 PM
Truthbetold
Nonlinear ODE
$\displaystyle y' = 4t^2y^2$

Hint: write the ODE as (F(y))' = G(t).

I don't understand how to use the hint. I understand that I need to combine y' and y^2 into the form (F(y))'; I just don't know how.

I only got to $\displaystyle \frac{y'}{y^2} = 4t^2$

Thanks!
• Jan 19th 2011, 07:22 PM
Prove It
$\displaystyle \displaystyle \frac{dy}{dt} = 4t^2y^2$

$\displaystyle \displaystyle y^{-2}\,\frac{dy}{dt} = 4t^2$

$\displaystyle \displaystyle \int{y^{-2}\,\frac{dy}{dt}\,dt} = \int{4t^2\,dt}$

$\displaystyle \displaystyle \int{y^{-2}\,dy} = \int{4t^2\,dt}$.

Go from here.
• Jan 20th 2011, 04:21 AM
Jester
Quote:

Originally Posted by Truthbetold
$\displaystyle y' = 4t^2y^2$

Hint: write the ODE as (F(y))' = G(t).

I don't understand how to use the hint. I understand that I need to combine y' and y^2 into the form (F(y))'; I just don't know how.

I only got to $\displaystyle \frac{y'}{y^2} = 4t^2$

Thanks!

Notice that $\displaystyle \dfrac{y'}{y^2} = - \dfrac{d}{dt} \left( \dfrac{1}{y}\right).$