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How did they get this? PDE

  1. Nov 2, 2013 #1
    I've got this far on a pde (second last step) but have no idea how they got this equality(I'm a noob), could someone please explain? I was going to put this under homework but it is not homework and it doesn't really fit the template. Thanks in advance.
     

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  3. Nov 2, 2013 #2

    arildno

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    It is an ordinary diff.eq,
    where dT/dt=-k*T, for a "k" with fluffy garments.
     
  4. Nov 3, 2013 #3

    HallsofIvy

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    You can see that [itex]T(t)= Ae^{(ih/E)t}[/itex] does satisfy the equation by differentiating. As arildno says, you can just think of k= (ih/E) and use the fact that the derivtive of [itex]Ae^{kt}[/itex] is [itex]kAe^{kt}[/itex].

    If you are asking "how can we get that solution if we didn't notice that?", you need to integrate to go the other way:
    [tex]\dfrac{dT}{dt}= -\dfrac{ih}{E}T[/tex]
    separating variables,
    [tex]\dfrac{dT}{T}= -\dfrac{ih}{E}dt[/tex]
    [tex]\int\dfrac{dT}{T}= -\dfrac{ih}{E}dt[/tex]
    [tex]ln(T)= -\dfrac{ih}{E}t+ C[/tex]

    Now, take the exponential of both sides:
    [tex]T= e^{-\frac{ih}{E}t}e^C[/tex]
    and we let [itex]A= e^C[/itex].
     
  5. Nov 3, 2013 #4
    Thanksbut I got it a while ago now. sorry for not mentioning.
     
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