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Thread: second order x function 14

  1. #1
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    second order x function 14

    x^2y''+xy'-0.25=0
    how to solve it?
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  2. #2
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    $\displaystyle \displaystyle x^2\,\frac{d^2y}{dx^2}+ x\,\frac{dy}{dx} - \frac{1}{4} = 0$

    $\displaystyle \displaystyle \frac{d^2y}{dx^2} + \frac{1}{x}\,\frac{dy}{dx} - \frac{1}{4x^2} = 0$

    $\displaystyle \displaystyle \frac{d^2y}{dx^2} + \frac{1}{x}\,\frac{dy}{dx} = \frac{1}{4x^2}$.


    Now let $\displaystyle \displaystyle u = \frac{dy}{dx}$ so that $\displaystyle \displaystyle \frac{du}{dx} = \frac{d^2y}{dx^2}$ and the DE becomes

    $\displaystyle \displaystyle \frac{du}{dx} + \frac{1}{x}\,u = \frac{1}{4x^2}$.

    This is first order linear, so use the Integrating Factor method to find $\displaystyle \displaystyle u$, which you can then use to find $\displaystyle \displaystyle y$.
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