Diff. eq.

Discussion in 'Math' started by boks, Nov 27, 2008.

  1. boks

    Thread Starter Active Member

    Oct 10, 2008
    218
    0
    1. The problem statement, all variables and given/known data

    The function u(x,t) satisfies the equation

    (1) u_{xx} = u_{tt} for 0 < x < pi, t > 0

    and the boundary conditions

    (2) u_x(0,t) = u_x(pi, t) = 0

    Show that (1) and (2) satisfy the superposition principle.

    2. The attempt at a solution

    I let w(x,t) = au(x,t) + bv(x,t) for two constants a and b.

    w_{tt} = au_{tt} + bv_{tt} = au_{xx} + bv_{xx} = cw_{xx}, where c is a constant

    Have I now showed that w(x,t) satisfies (1)? w_{xx} is not equal to w_{tt} unless c is 1...
     
    Last edited: Nov 27, 2008
  2. blazedaces

    Active Member

    Jul 24, 2008
    130
    0
    You have not shown that w(x,t) satisfies (1). May I ask what "w_{xx} is not equal to w_{tt} unless c is 1..." have to do with the superposition principle?

    The part that is going about it correctly is when you do the following: "au_{tt} + bv_{tt} = au_{xx} + bv_{xx}". But you need to prove that to be true, you can't just write it down...

    By the way, are you studying waves at the moment, because part 1 is part of the definition of a wave...

    -blazed
     
    Last edited: Nov 27, 2008
  3. boks

    Thread Starter Active Member

    Oct 10, 2008
    218
    0
    I figured it out, thanks.

    I'm studying PDEs.
     
  4. blazedaces

    Active Member

    Jul 24, 2008
    130
    0
    I see. That's good.

    Cheers,

    -blazed
     
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