1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Differential Equations model

  1. Oct 8, 2013 #1
    1. The problem statement, all variables and given/known data

    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?

    2. Relevant equations

    3. 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?

    [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|

    [itex]\pm[/itex]e^(-kt)*e^(-c) =M-P
    [itex]\pm[/itex] e^(-c) is a constant so call it A
    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?
  2. jcsd
  3. Oct 9, 2013 #2
    So then it would be: let -A=D

  4. Oct 9, 2013 #3


    Staff: Mentor

    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Threads - Differential Equations model Date
Modeling epidemics - solving differential equation Apr 22, 2017
Population growth with mice Mar 19, 2017
Modeling epidemics - solving differential equation Nov 19, 2016
Modeling differential equation Nov 19, 2016