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ThomasMagnus
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Homework Statement
Model for learning in the form of a differential equation:
[itex]\frac{dP}{dt}[/itex]= 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
[itex]\frac{dP}{M-P}[/itex] = kdt
[itex]\int\frac{dP}{M-P}[/itex] = k [itex]\int dt [/itex]
for [itex]\int\frac{dP}{M-P}[/itex] let u=M-P, du=-dP
-[itex]\int\frac{1}{u}[/itex] du= -ln|M-P|
-ln|M-P|=kt+C
ln|M-P|=-kt-c
e^(-kt-c)=|M-P|
[itex]\pm[/itex]e^(-kt)*e^(-c) =M-P
[itex]\pm[/itex] 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: [itex]\int\frac{dP}{P-M}[/itex] = [itex]\int -kdt [/itex] and get a final answer of: P(t)= M+Ae^(-kt). Are these answers equivalent because we can just say -A is another constant?