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Homework Help: Proving the superposition principle

  1. Jan 24, 2008 #1
    1. The problem statement, all variables and given/known data

    Hi everyone.
    I am trying to prove the superposition principle for linear homogeneous equations, which states that if u(t) and w(t) are solutions to y' + p(t)y = 0, then u(t) + w(t) and k(u(t)) are also solutions for any constant k.


    3. The attempt at a solution
    I substituted (u + w) in for y, but how does that help me?

    Thanks in advance for the help!
     
  2. jcsd
  3. Jan 24, 2008 #2
    Use the fact that [itex]u(t),w(t)[/itex] satisfy the ODE.
     
  4. Jan 24, 2008 #3
    As in, what differentiation property makes u + w a solution if u, w are?
     
  5. Jan 24, 2008 #4
    The statement [itex]u(t),w(t)[/itex]: solutions of the ODE means

    [tex]u'(t)+p(t)\,u(t)=0, \, w'(t)+p(t)\,w(t)=0[/tex]

    Put [itex]y'(t)=u'(t)+w'(t)[/itex] in the original ODE and use the above equations.
     
  6. Jan 24, 2008 #5
    Hah, I got it. That wasn't bad at all.
    Thanks!
     
  7. Jan 25, 2008 #6

    HallsofIvy

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    Science Advisor

    Now, can you do it for ku(t)? And, can you prove that those two together show that
    au(t)+ bv(t) is a solution for any numbers a, b, as long as u(t) and v(t) are solutions?
     
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