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Math Help - differential equations

  1. #1
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    differential equations

    dy/dx - 2y = 4x^2

    with y(0) = -1

    any help would be appreciated
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  2. #2
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    Quote Originally Posted by bobguy View Post
    dy/dx - 2y = 4x^2

    with y(0) = -1

    any help would be appreciated
    This is first order linear, so it is solved using the Integrating Factor.

    The integrating factor is

    e^{\int{-2\,dx}} = e^{-2x}.


    Multiply both sides by the integrating factor:

    e^{-2x}\frac{dy}{dx} - 2e^{-2x}y = 4x^2e^{-2x}

    \frac{d}{dx}\left(e^{-2x}y\right) = 4x^2e^{-2x}

    e^{-2x}y = \int{4x^2e^{-2x}\,dx}

    Now evaluate the right hand side using integration by parts (twice).
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