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Linear Partial Differential Equation-Problem in solving?

  1. Sep 10, 2006 #1
    Linear Partial Differential Equation---Problem in solving??

    Hi guys!

    im getting stuck in solving the following Linear Partial Differential Equation:


    now when i just form its auxiliary equation it becomes

    dx/2 = dy/3 = dz/(x+y+1)

    Now the problem that im just facing is how to solve these three sides. Usually i try to divide and multiply by some constant and then add these three. But in this case, i just couldnt find a way to go!

    please help me as soon as possible!

    thanks in advance!
  2. jcsd
  3. Sep 14, 2006 #2
    You are trying to solve linear first order PDE by the method of characteristics and this way leads certainly to the right answer. However there exist many different methods for solving the problem. I would like to point out the uncommon method - operator method.

    If rewrite your equation in the following form

    dz/dx =- 3/2*dz/dy + f(x,y)

    where f(x,y)=(x+y+1)/2

    then its solution in the operator form (See http://arxiv.org/abs/math-ph/0409035 ) follows immediately as

    z(x,y) = exp(-x*(3/2)*d/dy)*F(y)+int_0^x[exp(-(x-t)*(3/2)*d/dy)* f(t,y)] dt
    where F(y) is an arbitrary function.

    Since the operator exp(-x*(3/2)*d/dy) here is the simple shift operator we arrive to

    z(x,y) = F(y-x*(3/2))+int_0^x[f(t,y-(x-t)*(3/2))] dt

    =F(y-x*(3/2))+x^2/8+xy/2+x/2 .

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