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Math Help - Checking differential equation

  1. #1
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    Checking differential equation

    Can someone check my working?

    Solve xy'=2x^2y+y\ln y by letting \ln y=v

    \ln y=v\Rightarrow y=e^v

    \dfrac{dy}{dx}=\dfrac{dy}{dv}\cdot\dfrac{dv}{dx}

    \dfrac{dy}{dx}=e^v \dfrac{dv}{dx}

    xe^v \dfrac{dv}{dx}=2x^2e^v+ve^v

    Divide by e^v

    x \dfrac{dv}{dx}=2x^2+v

    \dfrac{dv}{dx}=2x+\dfrac{v}{x}

    \dfrac{dv}{dx}-\dfrac{v}{x}=2x

    I=e^{\int{-\frac{1}{x}} \ dx}=\dfrac{1}{x}

    \dfrac{1}{x} \dfrac{dv}{dx}-\dfrac{v}{x^2}=2

    \left[\dfrac{1}{x}v\right]'=2

    \dfrac{1}{x}v=2x+C

    v=2x^2+Cx

    \ln y =2x^2+Cx

    \Rightarrow y=e^{2x^2+Cx}
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  2. #2
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    It all looks good to me...
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