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Math Help - partial differential equation

  1. #1
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    Lightbulb partial differential equation

    Let u = u(x,y) be the complete integral of the PDE

      \frac {\partial u} {\partial x} . \frac {\partial u} {\partial y} = xy passing through the points
    (0,0,1) and (0,1,1/2) in the x-y-u space.

    Compute the value of u(x,y) evaluated at (-1,1) is _____

    I'm completely lost with respect to this problem.. 'd be great if someone can show me the right direction.


    Thanks,
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  2. #2
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    Re: partial differential equation

    \displaystyle \begin{align*} \\ \frac {\partial u} {\partial x} . \frac {\partial u} {\partial y}\ &=\ xy &=\ (kx) (\frac{y}{k})\ & \Leftarrow\ \LARGE{[}\ \frac {\partial u} {\partial x} = kx \And \frac {\partial u} {\partial y} = \frac{y}{k}\ \Huge{]} \end{align*}


    Spoiler:

    Just in case a picture helps...




    Spoiler:

    \displaystyle \begin{align*} \text{E.g.,}\ \ u(0,0)\ =\ 1\ =\ \frac{k}{2}(0)^2 + \frac{1}{2k}(0)^2 + c\ \Rightarrow\ c = 1 \\ \\ \text{And, e.g.,}\ \ u(0,1)\ =\ \frac{1}{2}\ =\ \frac{k}{2}(0)^2 + \frac{1}{2k}(1)^2 + 1\ \Rightarrow\ \text{etc.} \end{align*}




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  3. #3
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    Re: partial differential equation

    Tom, thanks for the reply! that definitely showed a lot of light on how to get started !!! Thanks again, TOM@ballooncalculus... The image was thoughtfully designed...
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