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Math Help - 2nd order PDE- general solution

  1. #1
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    2nd order PDE- general solution

    Hi all,

    I am trying to determine the general solution for the following equation:



    First of all i factorised it and determined the 1st order ODEs to be:



    For equation (1) there are no characterisitcs. While, for equation (2), i determined the characterisitics to be c_2 = x - t

    The answer is.... the general solution,
    phi(x,t) = F(t) + G(x -t)

    Could some explain to me why there is F(t) in the general solution and why is it a function of t?

    Thanks in advance,
    ArTiCk
    Attached Thumbnails Attached Thumbnails 2nd order PDE- general solution-equation.jpg   2nd order PDE- general solution-equation-1.jpg  
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  2. #2
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    Well, if \phi(x,t)=F(t), then \frac{\partial \phi}{\partial x}=0.

    \frac{\partial \phi}{\partial x} kills of all functions of t.
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  3. #3
    MHF Contributor

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    Quote Originally Posted by ArTiCK View Post
    Hi all,

    I am trying to determine the general solution for the following equation:



    First of all i factorised it and determined the 1st order ODEs to be:

    This is simply wrong. You cannot "factor" derivatives like that.
    You can write it as \frac{\partial}{\partial x}\left(\frac{\partial \phi}{\partial x}+ \frac{\partial \phi}{\partial t}\right)= 0 which says that \frac{\partial \phi}{\partial x}+ \frac{\partial \phi}{\partial t} is independent of x. That is, that \frac{\partial \phi}{\partial x}+ \frac{\partial \phi}{\partial t}= F(t) where F(t) can be any function of t only.

    For equation (1) there are no characterisitcs. While, for equation (2), i determined the characterisitics to be c_2 = x - t

    The answer is.... the general solution,
    phi(x,t) = F(t) + G(x -t)

    Could some explain to me why there is F(t) in the general solution and why is it a function of t?

    Thanks in advance,
    ArTiCk
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