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Math Help - Nonlinear ODE

  1. #1
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    Nonlinear ODE

    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!
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  2. #2
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    \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}.


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  3. #3
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    Quote Originally Posted by Truthbetold View Post
    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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