Verifying and ploting D.E. solution

  • Thread starter morrobay
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In summary, the conversation discusses a homework problem involving a cup of coffee cooling according to the equation dT/dt = .17(36-T). The initial temperature is given as 85 degrees Celsius at t=0. The conversation includes two questions about verifying the solution and plotting a continuous curve. The solution is shown to be T(t) = 36 + Ce^-.17t and the conversation also mentions using numerical solutions and integrating both sides of the equation. Eventually, the conversation verifies the solution and clarifies a possible mistake in the equation.
  • #1
morrobay
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Homework Statement



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

Homework 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?

Homework Statement

 
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  • #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.
 
  • #3
slider142 said:
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.

Thanks.
Just for my edification would you or someone else complete the numerical integration on this problem. ( it is not a homework problem )
 
  • #4
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.
 
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  • #5
Thanks that is what I was looking for
 
  • #6
HallsofIvy said:
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.

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:

Related to Verifying and ploting D.E. solution

1. What is the purpose of verifying and plotting D.E. solutions?

The purpose of verifying and plotting differential equation (D.E.) solutions is to ensure the accuracy and validity of the solution. It allows us to check if the solution satisfies the original equation and to visualize the behavior of the solution over time.

2. How do you verify a D.E. solution?

To verify a D.E. solution, we substitute the solution into the original equation and simplify to see if both sides are equal. If they are, then the solution is considered verified.

3. What is the significance of plotting a D.E. solution?

Plotting a D.E. solution helps us understand the behavior of the solution over time and see how it changes with different initial conditions. It also allows us to identify any critical points or asymptotes.

4. What are some common methods for plotting D.E. solutions?

Some common methods for plotting D.E. solutions include using a graphing calculator, creating a table of values and graphing them by hand, or using a computer program such as MATLAB or Wolfram Alpha.

5. Can you verify and plot any type of D.E. solution?

Yes, the process of verifying and plotting D.E. solutions applies to all types of differential equations, including first-order, second-order, and systems of equations. However, the methods and techniques may vary depending on the type of equation and its complexity.

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