Trouble with Solving a Differential Equation

  • Thread starter xxdrossxx
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I'm having trouble with the following textbook problem:

A model for learning in the form of a differential equation is seen below:
[tex]\frac{dP}{dt} = k(M - P)[/tex]
where [tex]P(t)[/tex] measures the performance of someone learning a skill after a training time [tex]t[/tex], [tex]M[/tex] is the maximum level of performance, and [tex]k[/tex] is a positive constant. Solve this differential equation for [tex]P(t)[/tex]. What is the limit of this expression?

The only basic approach for solving differential equations that I know is to "separate and integrate", that is, have one side with P's and a dP, and the other side with t's and a dt (constants anywhere throughout), and then integrate both sides. So, what I was hoping to get was kM * something * dt = kP * something * dP. However, it looks like that's not going to happen. Could someone help me solve this thing?

I know that if "M is the maximum level of performance", then logically [tex]\lim_{t \rightarrow \infty} P(t) = M[/tex]. However, I'm still not sure how to actually find the function P(t).

Thanks for any help you can give. :smile:
 
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  • #2
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xxdrossxx said:
I'm having trouble with the following textbook problem:



The only basic approach for solving differential equations that I know is to "separate and integrate", that is, have one side with P's and a dP, and the other side with t's and a dt (constants anywhere throughout), and then integrate both sides. So, what I was hoping to get was kM * something * dt = kP * something * dP. However, it looks like that's not going to happen. Could someone help me solve this thing?

I know that if "M is the maximum level of performance", then logically [tex]\lim_{t \rightarrow \infty} P(t) = M[/tex]. However, I'm still not sure how to actually find the function P(t).

Thanks for any help you can give. :smile:


It's easily separable just divide by by M-P and you have a pretty standard integral to find the function P.
 
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d_leet said:
It's easily separable just divide by by M-P and you have a pretty standard integral to find the function P.
Ah, of course! I guess my first mistake was distributing the k. Dividing brings me to [tex]\frac{dP}{M - P} = kdt[/tex], and integrating both sides gives me [tex]-ln|M - P| = kt + c[/tex]. Going from there:

[tex]ln|M - P| = -kt - c[/tex]
[tex]e^{-kt - c} = M - P[/tex]
[tex]P = M - e^{-kt - c}[/tex]
I think I'm doing something very wrong here. The answer in the back of the book is [tex]P(t) = M - Me^{-kt}[/tex].

I'm thinking I've made a really stupid mistake, but I can't find it. Can anyone help?
 
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  • #4
HallsofIvy
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Have you left out part of the problem- like, say, the information that the subject knew nothing to start with? That is, that P(0)= 0.

[tex]P(t) = M - e^{-kt - c}= M- (e^{-c})e^{-kt}[/tex]
and we can write that as
[tex]P(t)= M- C e^{-kt}[/tex]
where [itex] C= e^{-c}[/itex].

Then, if P(0)= 0,
[tex]P(0)= M- Ce^0= M- C= 0[/tex]
so C= M.

Actually that's not relevant to the final question: what is the limit of P(t) as t goes to infinity? That answer does not depend on C.
 

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