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Math Help - partial differential equation, complete integral

  1. #1
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    Lightbulb partial differential equation, complete integral

    This is my question.

    Find the complete integral of the PDE

     \frac {\partial ^2 u}{\partial x^2} +2 \frac {\partial ^2 u}{\partial x \partial y}+ \frac{\partial ^2 u}{\partial y^2} = x ex+y

    involving arbitrary functions f1 and f2

    A little nudge towards the right direction would be of great help. I'm sure I can take it from there...
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  2. #2
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    Re: partial differential equation, complete integral

    Undetermined coefficients gets you there doesn't it ?
    Assume that
    u(x,y) = f(x)e^{x+y},
    substitute, and deduce a second order linear ode for f(x).
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  3. #3
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    Re: partial differential equation, complete integral

    Thank you BobP.

    I could get as far as factoring the auxiliary equation and getting two equal roots as 1, and -1.
    I guess the solution should be of the form (A+Bx)e -x.

    But i'm not able to get the next step to solve for the complete integral.
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  4. #4
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    Re: partial differential equation, complete integral

    I think that your Auxiliary Equation is wrong, I think it should be m^{2}+4m + 4 = 0
    having equal roots -2,-2.
    For the particular integral, you can use the undetermined coefficients method, let f(x)=Ax+B.
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  5. #5
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    Re: partial differential equation, complete integral

    BobP, can you post a link from where I can review my notes on finding the auxillary equation for such
    partial differential equations? That'd be of great help as I can improvize and find the right set of solutions...

    Thanks,
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  6. #6
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    Re: partial differential equation, complete integral

    Hi !
    The three main steps for solving the PDE are summarized in attachment (without the whole calculus, that you certainly can do by yourself).
    Attached Thumbnails Attached Thumbnails partial differential equation, complete integral-pde.jpg  
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  7. #7
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    Re: partial differential equation, complete integral

    Hi JJacquelin,

    I solved the homogenous equation and got -1and -1 as the two roots.

    Does this mean that the complementary function of the pde is,

     \phi_{1}(y-x) +x \phi_{2}(y-x) ?

    I guess it could also be


     \phi_{1}(x-y) +x \phi_{2}(x-y)

    As far as the particular integral is concerned, I'm a little at sea when i try following the image file you had attached with. Wish I was smart enough to get the gist of what you meant to explain..
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  8. #8
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    Re: partial differential equation, complete integral

    F(y-x) is solution of the homogeneous equation.
    But since F is any function, F(y-x) is the same as G(x-y) with any function G , and the same as H(exp(y-x)) with any function H , and the same as... many others.
    So you can chose any one of these functions containing (y-x). The function F(x-y)=F(-(y-x)) is one of them.
    Now, to find a particular solution of the complete PDE, the method is already given in the preceeding post (part 2). Just apply it : bring back u=(ax+b)exp(x+y) into the PDE.
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