Nonlinear ODE

**Truthbetold** · January 19th 2011, 07:15 PM

$y' = 4t^2y^2$

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 $\frac{y'}{y^2} = 4t^2$

Thanks!

**Prove It** · January 19th 2011, 07:22 PM

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

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

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

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

Go from here.

**Danny** · January 20th 2011, 04:21 AM

Originally Posted by Truthbetold

$y' = 4t^2y^2$

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 $\frac{y'}{y^2} = 4t^2$

Thanks!

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

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