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Math Help - Determine the general solution of this QL PDE

  1. #1
    Senior Member bugatti79's Avatar
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    Determine the general solution of this QL PDE

    1)Determine the general solution of the following QL PDE

    2) Use implicit differentiation to verify the solution.

    u u_x-y u_y=y

    My attempt at 1)

    \frac{dx}{u}=\frac{dy}{-y}=\frac{du}{y}

    Take the second two

    \int-dy=\int du \implies u=-y+A

    Taking the first two

    \frac{dx}{(-y+A)}=\frac{dy}{-y} \implies dx=\frac{(-y+A)dy}{-y}

    Integrating gives

    x=y-A \ln(y) + f(A) but f(A)=u+y

    therefore the general solution implicitly is

    x=y-A \ln(y) + f(u+y)

    How am I doing?

    Assuming the baove is right, how do I attempt 2)?

    Please note I have this posted in the physics help forum at

    Determine the general solution of QL PDE

    I will keep each forum informed of the other to ensure no one's time is wasted.

    Thanks
    Last edited by bugatti79; August 7th 2011 at 04:27 AM. Reason: Left out function f
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  2. #2
    MHF Contributor
    Jester's Avatar
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    Re: Determine the general solution of this QL PDE

    Now replace A with u + y so

    x = y - (u+y) \ln y + f(u + y).

    Now differentiate

    1 = -u_x \ln y + f'(u+y) u_x

    0 = 1 - (u_y + 1) \ln y - \frac{u+y}{y} + f'(u+y)(u_y+1)

    Solve for u_x and u_y and sub. into  u u_x - yu_y and simplify.
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  3. #3
    Senior Member bugatti79's Avatar
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    Re: Determine the general solution of this QL PDE

    Nice one Danny!! Thanks :-)
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