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Math Help - find the integrating factor and then use it to derive the general solution y(x)

  1. #1
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    find the integrating factor and then use it to derive the general solution y(x)

    Find the integrating factor of the equation: [e^y*x^2*cosx+1]dx+[e^y*x^2*sinx]dy=0 and use the integrating factor to find the general solution. Please help!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Hint : The equation has an integrating factor that depends on x .
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    So if I let M=e^y*x^2*cosx+1, then M with respect to y is x^2*e^y*cosx+y
    If N=e^y*x^2*sinx, then N with respect to x is e^y(2xsinx-x^2+2cosx) right?

    So if I use (Ms of y-Ns of x)/N=((x^2*e^y*cosx+y)-(e^y(2xsinx-x^2+2cosx))/(e^y*x^2*sinx).

    But it's so hard to simplify this to get the integrating factor. Or am I doing something wrong?
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    We have

    ( 1 / N ) ( M_y - N_x ) = ... = - 2 / x

    so, the integrating factor m ( x ) satisfies

    m ' ( x ) / m ( x ) = - 2 / x
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  5. #5
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    wait, so you got -2/x after dividing (M_y-N_x) by N?
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Taurus3 View Post
    wait, so you got -2/x after dividing (M_y-N_x) by N?

    And you should get the same.
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