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Proof on Linear 1st Order IVP solution being bounded

  1. Aug 21, 2013 #1
    Proof on Linear 1st Order IVP solution being "bounded"

    A function h(t) is called "bounded" for t≥t0 if there is a constant M>0 such that

    |h(t)|≤M for all t≥0

    The constant M is called a bound for h(t). Consider the IVP

    x'=-x+q(t), x(0)=x0

    where the nonhomogeneous term q(t) is bounded for t≥0. Show the solution of this IVP is bounded for t≥0. (Hint: Use the Variation of Constants Formula.)

    Any help on where to go for this problem would be great. Thanks
     
  2. jcsd
  3. Aug 22, 2013 #2

    HallsofIvy

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    You are given the "Hint: use the Variation of Constants Formula". Okay, what is that formula?
     
  4. Aug 22, 2013 #3
    The Variation of Constants formula is a generalized formula for First Order Linear DE's that can be solved with the Integrating Factor Method.

    I would put the exact formula down but I am not too familiar with this equation editor.
     
  5. Aug 22, 2013 #4

    LCKurtz

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    You have to show some effort. Show us what you have tried. I would give a second hint: It is a linear equation.
     
  6. Aug 25, 2013 #5
    Here is what I have tried:

    The Variation of Constants formula gave me this

    x=(x0+∫e^u q(u) du) e^-t

    the integral is definite and goes from 0 to t.

    Since q(t) is bounded, would that remain true if the integral is taken from it?
     
  7. Aug 25, 2013 #6

    LCKurtz

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    To rephrase your question, if you take the absolute value of both sides of that equation, can you overestimate the right hand side by some constant. So try it and show us what happens.
     
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