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Math Help - [SOLVED] Differentiation

  1. #1
    Member ronaldo_07's Avatar
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    [SOLVED] Differentiation

    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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    Quote Originally Posted by ronaldo_07 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?
    Well, how about using the hint? That equation is the same as
    (e^y+ e^{-x}ln|x|)dx+ (e^y+ y^2e^{-x})dy= 0 and you are told that e^x is an integrating factor. That means that if you multiply by e^x on both sides,
    (e^{x+y}+ ln|x|)dx+ (e^{x+y}+ y^2)dy= 0 is an "exact equation". Find a function F(x,y) such that Fx= e^{x+y}+ ln|x| and Fy= e^{x+y}+ y^2.
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