Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

First order linear partial differential equation

  1. Feb 26, 2009 #1
    Do these equations have two general solutions!?

    e.g. z_x + z_y -z = 0

    Using the method of characteristics

    a=1
    b=1
    c=-1
    d=0

    Therefore dx/1=dy/1=dz/z

    Taking first two terms: x = y + A
    *Taking last two terms: z = Be^y
    So general solution is z = f(x-y)e^y

    BUT if we took first and last terms: z=Be^x
    z=f(x-y)e^x.......
     
  2. jcsd
  3. Feb 26, 2009 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Neither of those is the "general" solution. z= f(x-y)ex+ g(x-y)ey is the general solution.
     
  4. Feb 27, 2009 #3
    You are quite the genius! Thanks
     
  5. Apr 2, 2009 #4
    I disagree - first order PDE's don't have two arbitrary functions in their solutions!

    Actually, they're both right.

    First solution [tex]z = e^xf(x-y)[/tex] second solution [tex]z = e^y g(x-y)[/tex]. Since [tex]f[/tex] is arbitrary the set [tex]f(x-y) = e^{-(x-y)} g(x-y)[/tex] and the first becomes the second.
     
  6. Apr 3, 2009 #5

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    What, you mean I'm NOT a genius?
     
  7. Apr 3, 2009 #6
    I've never met you so I really don't know :rofl:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: First order linear partial differential equation
Loading...