1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    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!

Homework Help: Differential Equations Problem

  1. Mar 17, 2013 #1
    1. The problem statement, all variables and given/known data
    This is actually a question we went over in class, but I kind of spaced out when my teacher was explaining it. I have since solved it myself while I was reviewing for my calculus test, but when I compared my answer with the answer key my teacher provided to our class, I found out my answer was wrong. Here's the question.

    "Write and solve the differential equation that models the statement in the following problem.

    The rate of change of P with respect to t is proportional to 10-t."

    2. Relevant equations
    I should add that this is part of our chapter involving exponential growth and decay.

    3. The attempt at a solution
    Now, I understand how to do this particular problem. You essentially start off with the following differential equation:

    dP/dt = K/(10-t)

    You then multiply the dt over to isolate all the t variable expressions on the right side.

    dP = [K/(10-t)]dt

    Then you would take an indefinite integral of both sides respectively. The problem is, my teacher says that the answer should be:

    P = -K*ln|10-t| + C

    When I solved this on my own, I produced the same result essentially, just without a negative sign in front of the K constant. Am I missing something here? Could someone please explain to me why there should be a negative sign or did my teacher make a mistake?
  2. jcsd
  3. Mar 17, 2013 #2


    User Avatar
    Homework Helper

    Did you integrate the right side using u-substitution? After you use that method, the negative will appear in the answer.
  4. Mar 17, 2013 #3
    Ah, no. I didn't. I tried integrating using substitution just now, and it worked. >.<
    Thank you for pointing that out to me. It makes sense now.
  5. Mar 18, 2013 #4


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    When you say ##y## is proportional to ##x## that means there is a constant ##k## such that ##y=kx##. So if the rate of change of ##P## is proportional to ##10-t##, then ##\frac{dP}{dt}= k(10-t)##. So you either are solving the problem wrong or stated the problem you are solving wrong.
    Last edited: Mar 18, 2013
  6. Mar 18, 2013 #5
    That's my mistake. I stated the problem incorrectly. It should say "inversely proportional". But regardless, eumyang pointed out what I was missing. I didn't use the substitution method to integrate the differential equation, so I never produced that negative sign.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted