Results 1 to 3 of 3

Math Help - Exact Equation

  1. #1
    Newbie staevobr's Avatar
    Joined
    Sep 2010
    Posts
    10

    Exact Equation

    (e^x \sin{y} -3x^2) \,dx +(e^x \cos{y} + \frac{y^{-2/3}}{3}) \, dy = 0

    (e^x \sin{y} -3x^2) +(e^x \cos{y} + \frac{y^{-2/3}}{3}) \frac{dy}{dx} = 0<br />

    M = e^x \sin{y} -3x^2 \,, N = e^x \cos{y} + \frac{y^{-2/3}}{3}

    \frac{\partial M}{\partial y} = e^x \cos{y}

    \frac{\partial N}{\partial x} = e^x \cos{y}

    Since
     \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x},
    there exists an F such that
    \frac{d}{dx}F(x, y) = 0

    F = \int{M dx}
    F = \int{e^x \sin{y} -3x^2 dx} = e^x \sin{y} -x^3 + h(y)

    \frac{\partial F}{\partial y} = N

    e^x \cos{y} + \frac{y^{-2/3}}{3} = e^x \sin{y} -x^3 + h'(y)

    I can't find any way to get past this point. Is there any possible function h'(y) that would satisfy this equality? Or have I made an error?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,492
    Thanks
    1393
    Your working is a little off...

    Since \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}

    that means

    \frac{\partial F}{\partial x} = M and \frac{\partial F}{\partial y} = N.

    So

    \frac{\partial F}{\partial x} = e^x\sin{y} - 3x^2

    F = \int{e^{x}\sin{y} - 3x^2\,dx}

     = e^x\sin{y} - x^3 + g(y).


    \frac{\partial F}{\partial y} = e^x\cos{y} + \frac{y^{-\frac{2}{3}}}{3}

    F = \int{e^x\cos{y} + \frac{y^{-\frac{2}{3}}}{3}\,dy}

     = e^x\sin{y} + y^{\frac{1}{3}} + h(x).


    So we have at the same time...

    F = e^x\sin{y} - x^3 + g(y) and F = e^x\sin{y} + y^{\frac{1}{3}} + h(x).

    This means g(y) = y^{\frac{1}{3}} + C_1 and h(x) = -x^3 + C_2.


    Putting everything together gives

    F = e^x\sin{y} - x^3 + y^{\frac{1}{3}} + C where C = C_1 + C_2.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie staevobr's Avatar
    Joined
    Sep 2010
    Posts
    10
    Thanks.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 5
    Last Post: September 21st 2011, 12:12 AM
  2. Exact equation
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: May 23rd 2011, 09:44 AM
  3. non exact equation
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: November 23rd 2010, 03:30 AM
  4. Exact equation
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: February 9th 2010, 03:04 PM
  5. Exact Solution of the Equation
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: November 11th 2009, 08:20 AM

/mathhelpforum @mathhelpforum