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Unique solution of 1st order autonomous, homogeneous DE

  1. Sep 20, 2010 #1
    Hello,

    1st order autonomous, homogeneous differential equation have the general form:
    [tex]\dot{x}(t)=ax(t)[/tex]

    It can be shown that the unique solution is always:

    [tex]x(t)=e^{at}x(t_{0})[/tex]

    I don't get this, could anyone help me?

    Thanks!
     
  2. jcsd
  3. Sep 20, 2010 #2

    HallsofIvy

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    Have you tried at all? What is the derivative of [itex]e^{at}[/itex]. Since [itex]x(t_0)[/itex] is a constant, what is the derivative of [itex]e^{at}x(t_0)[/itex]?
     
  4. Sep 20, 2010 #3
    Thank you for your answer.

    I can work it out when x(t) = x, but this is not the case, is it?
     
  5. Sep 20, 2010 #4

    Office_Shredder

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    x(t) is supposed to be a function of t. What do you mean by x(t)=x?
     
  6. Sep 20, 2010 #5
    How can I calculate the integral of x(t) when I don't know the corresponding function? x(t) can equal (t^2) or (t-3) and so on, right?
     
  7. Sep 21, 2010 #6

    HallsofIvy

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    Will you please answer my questions? Do you know what the derivative of [itex]e^{at}[/itex] is? Do you know what the derivative of [itex]e^{at}x(t_0)[/itex] is?
     
  8. Sep 21, 2010 #7

    HallsofIvy

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    You can't and you don't want to.

    If [tex]\frac{dx}{dt}= ax[/tex] then
    [tex]\frac{dx}{x}= adt[/tex]

    Integrate both sides of
    [tex]\int \frac{dx}{x}= \int a dt[/tex]

    Note that on the left you have NO "t". The only variable is x.
     
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