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Math Help - [SOLVED] Solve the following differential equation?

  1. #1
    Super Member fardeen_gen's Avatar
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    [SOLVED] Solve the following differential equation?

    Solve the following differential equation:
    3x^2y^2 + \cos (xy) - xy \sin (xy) + \frac{dy}{dx}\{2x^3y - x^2\sin (xy)\} = 0
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  2. #2
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    Quote Originally Posted by fardeen_gen View Post
    Solve the following differential equation:
    3x^2y^2 + \cos (xy) - xy \sin (xy) + \frac{dy}{dx}\{2x^3y - x^2\sin (xy)\} = 0
     \frac{dy}{dx}\{2x^3y - x^2\sin (xy)\} = xy \sin (xy) -3x^2y^2 - \cos (xy) +xy \sin (xy)

     \frac{dy}{dx} = \frac{xy \sin (xy) -3x^2y^2 - \cos (xy) }{2x^3y - x^2\sin (xy)}

     \int \frac{dy}{dx} dx = \int\frac{xy \sin (xy) -3x^2y^2 - \cos (xy) }{2x^3y - x^2\sin (xy)} dx

     \int dy = \int\frac{xy \sin (xy) -3x^2y^2 - \cos (xy) }{2x^3y - x^2\sin (xy)} dx

     y+c = \int\frac{xy \sin (xy) -3x^2y^2 - \cos (xy) }{2x^3y - x^2\sin (xy)} dx

    There's my lazy contribution!
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  3. #3
    Super Member fardeen_gen's Avatar
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    The solution is:
    Spoiler:
    x(x^2y^2 + \cos xy) = c

    I am unable to find it! Anyone?
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  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Quote Originally Posted by fardeen_gen View Post
    Solve the following differential equation:
    3x^2y^2 + \cos (xy) - xy \sin (xy) + \frac{dy}{dx}\{2x^3y - x^2\sin (xy)\} = 0
    Note that the differential equaiton is exact( you can check the partials) i.e there is a function z(x,y) such that

    dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy

    so

    \frac{\partial z}{\partial x} = 3x^2y^2 + \cos (xy) - xy \sin (xy)


    Now we take the partial integral with respect to x and get

    z=x^3y^2+\frac{\sin(xy)}{y}+x\cos(xy)-\frac{\sin(xy)}{y}+g(y)

    Now we take the partial with respect to y to get

    \frac{\partial z}{\partial y}=2x^3y-xy\sin(xy)+g'(y)

    Now we see that g'(y)=0

    So g(y)=c

    So we get an implict solution

    x^3y^2+x\cos(xy)=c
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