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Thread: Tricky First Order Differential Equation

  1. #1
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    Tricky First Order Differential Equation

    Plz help with this First Order Differential Equation.

    So the equation is: x*y'-y=x^3-1

    I can't separate the variables?

    Anyone knows how to solve that?
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  2. #2
    MHF Contributor chisigma's Avatar
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    The DE can be written in the form...

    y^{'}= a(x)\cdot y + b(x) (1)

    ... so that it is linear and can be solved in 'standard way'...

    Kind regards

    \chi \sigma
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  3. #3
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    Quote Originally Posted by gergofoto View Post
    Plz help with this First Order Differential Equation.

    So the equation is: x*y'-y=x^3-1

    I can't separate the variables?

    Anyone knows how to solve that?
    x\frac{dy}{dx} - y = x^3 - 1

    \frac{dy}{dx} - x^{-1}y = x^2 - x^{-1}.


    Now we need an integrating factor:

    e^{\int{-x^{-1}\,dx}} = e^{-\ln{x}} = e^{\ln{x^{-1}}} = x^{-1}.

    Multiply the DE through by the integrating factor:

    x^{-1}\frac{dy}{dx} - x^{-2}y = x - x^{-2}.

    Now the LHS is a product rule expansion of \frac{d}{dx}(x^{-1}y).

    So \frac{d}{dx}(x^{-1}y) = x - x^{-2}

    x^{-1}y = \int{x - x^{-2}\,dx}

    x^{-1}y = \frac{1}{2}x^2 + x^{-1} + C

    y = \frac{1}{2}x^3 + 1 + Cx.
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