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Diff. eq.

  1. Nov 27, 2008 #1
    1. The problem statement, all variables and given/known data

    The function u(x,t) satisfies the equation

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

    and the boundary conditions

    (2) [tex]u_x[/tex](0,t) = [tex]u_x[/tex](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.

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

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


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    Staff Emeritus
    Science Advisor

    You said you let w= au+ bv. What is c? What do you mean by "auxx+ bvxx= c wxx"? I don't see where that comes from.

    Perhaps it would be simpler to see if you rewrote the equation as uxx- utt= 0. What is wxx- wtt?
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