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Homework Help: PDE of a 3d function

  1. Jun 18, 2009 #1
    b]1. The problem statement, all variables and given/known data[/b]
    Find the characteristics, and then the solution, of the partial differential equation

    x[tex]\frac{\partial u}{\partial x}[/tex]+xy[tex]\frac{\partial u}{\partial y}[/tex]+z[tex]\frac{\partial u}{\partial z}[/tex]=0

    given that u(1, y, z)=yz

    2. Relevant equations

    3. The attempt at a solution

    Found this question on an old exam and am not quite sure what to do.
    Initialy i tried to take the z derivative to the other side and still take dy/dy=y to be the characteristics, leaving me to solve

    x[tex]\frac{\partial u}{\partial x}[/tex]+z[tex]\frac{\partial u}{\partial z}[/tex]=0.

    but i think thats wrong.
    Could someone please point me in the right direction.
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Jun 18, 2009 #2


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    Science Advisor

    Are you clear on what "characteristics" are for a partial differential equation?

    The characteristics for this equation are given by
    dx/dt= x, dy/dt= xy, and dz/dt= z. From the first and third, x= Aet and z= Cet. The second equation is dy/dt= Aety which separates as dy/y= Aetdt and integrates as ln(y)= Aet+ B' or y= BeAet= Bex. Also, z= Cet and, from x= Aet, et= x/A so z= Cex/A. That is, taking x= t as parameter, the charateristics are given by x= t, y= Bet, and z= Cet/A.
  4. Jun 18, 2009 #3
    Ok thanks. So would the solution of the PDE still be just an arbitary function of the constant combination of variables? how do i combine all the characteristics to form a single solution?
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