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Verifying and ploting D.E. solution

  1. Feb 7, 2009 #1

    morrobay

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    1. The problem statement, all variables and given/known data

    A cup of coffee cools according to;

    dT/dt = .17(36-T)
    At t=0 T= 85 deg centigrade
    Two questions : how to verify the solution below and to plot a continuous curve of T/t

    2. Relevant equations

    the solution is T(t) = 36 + Ce^-.17t, = 36+49e^-.17t


    b]3. The attempt at a solution[/b]

    I solved for two (t)'s numerically ;
    T= 74 , 1.29 deg=e^.17t , t= 1.5 s
    T= 80 ,.108=.17t . t=.64 s
    So could someone explain how to integrate this and get the curve?
    1. The problem statement, all variables and given/known data
     
  2. jcsd
  3. Feb 7, 2009 #2
    This equation is separable. Manipulate it algebraically so that you get the form f(T) dT/dt = g(t), then integrate both sides dt: [tex]\int f(T) \frac{dT}{dt} dt = \int g(t) dt[/tex]. Note that either f or g may be a constant function, and that some people are more comfortable writing [itex]\frac{dT}{dt} dt[/itex] as just dT.
     
  4. Feb 8, 2009 #3

    morrobay

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    Thanks.
    Just for my edification would you or someone else complete the numerical integration on this problem. ( it is not a homework problem )
     
  5. Feb 8, 2009 #4

    HallsofIvy

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    Why do you keep insisting on numerical solutions? It is easy to integrate both sides of
    [tex]\int\frac{dT}{36- T}= \int .17 dt[/tex]
    (On the left use the substitution u= 36- T.)

    Of course, to verify that [itex]T= 36+49e^{-.17t}[/itex] is the solution to the problem you don't even need to integrate. [tex]dT/dt= (-.17)(49)e^{-.17t}[/tex] while [tex]36- .17T= (-.17)(49)e^{-.17t}[/tex] so this T clearly satifies the equation and, of course, taking t= 0 gives T(0)= 36+ 49= 85.
     
    Last edited: Feb 8, 2009
  6. Feb 8, 2009 #5

    morrobay

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    Thanks that is what I was looking for
     
  7. Feb 9, 2009 #6

    morrobay

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    referring to : 36-.17T = (-.17)(49)e^-.17t
    It looks likes the left side above should be (36-T)(.17) since at (t)=1.5 T=74
    and the right side ,dT/dt= 6.45
     
    Last edited: Feb 9, 2009
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