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Math Help - Find the general solution to the equation 2u_x + u_xy = 0

  1. #1
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    Find the general solution to the equation 2u_x + u_xy = 0

    where u_x is u(x,y) differentiated with respect to x and u_xy is u(x,y) twice differentiated with respect to x and with respect to y

    How would I go about solving this?

    Thanks for any help.
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  2. #2
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    Re: Find the general solution to the equation 2u_x + u_xy = 0

    I'm pretty sure I've done it now actually.

    u = e^(ax+2y)
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  3. #3
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    Re: Find the general solution to the equation 2u_x + u_xy = 0

    What you present is just one solution. Here are two more

    (1) u = e^{-2y} \sin x

    (2) u = x e^{-2y} + y^2

    Why not let v = u_x and see where that gets you.
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  4. #4
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    Re: Find the general solution to the equation 2u_x + u_xy = 0

    Quote Originally Posted by Danny View Post

    Why not let v = u_x and see where that gets you.
    This gives 2v + v_y = 0

    I'm not really sure what to do
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  5. #5
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    Re: Find the general solution to the equation 2u_x + u_xy = 0

    Separate and integrate noting that you'll get a function of integration.
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  6. #6
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    Re: Find the general solution to the equation 2u_x + u_xy = 0

    Since only differentiation with respect to y is involved, you can solve that as the ODE
    \frac{dv}{dy}= -2y. Of course, the "constant" may be a function of x. What does that give you for v? What does that make the orginal equation?

    Remember that the general solution to a partial differential equation may involve unknown functions of the variables rather than unknown constants.
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  7. #7
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    Re: Find the general solution to the equation 2u_x + u_xy = 0

    Quote Originally Posted by HallsofIvy View Post
    Since only differentiation with respect to y is involved, you can solve that as the ODE
    \frac{dv}{dy}= -2y.
    surely you mean dv/dy = -2v ?
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  8. #8
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    Re: Find the general solution to the equation 2u_x + u_xy = 0

    Quote Originally Posted by feyomi View Post
    surely you mean dv/dy = -2v ?
    I'm sure that's what HoI meant.
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