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Math Help - [SOLVED] differential equation yrgent please

  1. #1
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    [SOLVED] differential equation yrgent please

    Use the integrating factor μ = ex to determine the general solution (in implicit form) of the differential equation

    e^y +e^{−x} ln |x| + (e^y + y^2e^−x)dy/dx=0

    How to solve this?
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  2. #2
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    Integrating factor

    Hello rajr
    Quote Originally Posted by rajr View Post
    Use the integrating factor μ = ex to determine the general solution (in implicit form) of the differential equation

    e^y +e^{−x} ln |x| + (e^y + y^2e^−x)dy/dx=0

    How to solve this?
    Multiply both sides by e^x:

    e^xe^y + \text{ln}|x| + (e^xe^y + y^2)\frac{dy}{dx}=0

    Now if we differentiate e^xe^y + \frac{1}{3}y^3 with respect to x, we get:

    e^xe^y\frac{dy}{dx} + e^xe^y + y^2\frac{dy}{dx}

    = e^xe^y + (e^xe^y + y^2)\frac{dy}{dx}

    So our equation is:

    \text{ln}|x| + \frac{d}{dx}(e^xe^y + \frac{1}{3}y^3) = 0

    \Rightarrow e^xe^y + \frac{1}{3}y^3 = -\int \text{ln}|x|

    \Rightarrow e^xe^y + \frac{1}{3}y^3 = -x \text{ln} |x| + x + c

    Grandad
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