How Do Differential Equations Model Learning Performance Over Time?

[ThomasMagnus]

Homework Statement

Model for learning in the form of a differential equation:

\frac{dP}{dt}= k(M-P)

Where P(t) measures the performance of someone learning a skill after training time (t), M is the maximum level of performance, and k is a positive constant. Solve this differential equation to find an expression for P(t). What is the limit of this expression?

Homework Equations

The Attempt at a Solution

I think I am doing this problem the correct way. However, my textbook uses a different method. Would you be able to confirm if I am do this correctly?

dP=k(M-P)dt
\frac{dP}{M-P} = kdt

\int\frac{dP}{M-P} = k \int dt

for \int\frac{dP}{M-P} let u=M-P, du=-dP

-\int\frac{1}{u} du= -ln|M-P|

-ln|M-P|=kt+C
ln|M-P|=-kt-c
e^(-kt-c)=|M-P|
\pme^(-kt)*e^(-c) =M-P
\pm e^(-c) is a constant so call it A
M-Ae^(-kt)=P(t)
as t→∞ P→M

The book does something different. At the very start they say: \int\frac{dP}{P-M} = \int -kdt and get a final answer of: P(t)= M+Ae^(-kt). Are these answers equivalent because we can just say -A is another constant?

 

[ThomasMagnus]

So then it would be: let -A=D

M+De^(-kt)=P(t)

 

[Mark44]

Per PF rules, you shouldn't bump a thread earlier than 24 hours after you post it. If you need to make a change, just edit your post.