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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.

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.

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