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!

Integrating factor

  1. Feb 23, 2016 #1

    EastWindBreaks

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    hello, I was reading through the text book and I have a hard time to understand this part:
    FullSizeRender (3).jpg

    2. Relevant equations


    3. The attempt at a solution
    haven't been dealing with derivatives for a while, i dont understand how it got ln |u(t)| from the first equation.
    Am I treating the derivative as a fraction here? how does u(t)/u(t) = ln |u(t)| ?

    any help is greatly appreciated
     
  2. jcsd
  3. Feb 23, 2016 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    The first equation is more usually written: $$\frac{1}{\mu}\frac{d\mu}{dt} = \frac{1}{2}$$ ... that help?
    The ##\ln|\mu|## part comes from the chain rule.

    I think the author is specifically trying not to treat the Leibnitz notation as a fraction, since the usual way to proceed from (1) would be to write: $$\frac{d\mu}{\mu} = \frac{dt}{2}$$ ... which is considered sloppy notation.
     
  4. Feb 23, 2016 #3

    Mark44

    Staff: Mentor

    "Sloppy" is in the eye of the beholder. The differential equation in the first paragraph above is separable, as can be seen in the equation immediately above. Separation of variables is a standard technique in solving differential equations. In this technique, which is one of the first taught in a course on ODE, derivatives in Leibniz form are treated as fractions.

    The next step is to integrate both sides, which yields ##\ln|\mu| = \frac 1 2 t + C##.
     
    Last edited: Feb 23, 2016
  5. Feb 25, 2016 #4

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    It's the way that produces the fewer headaches.
     
  6. Feb 25, 2016 #5

    Mark44

    Staff: Mentor

    At the very least, that's debatable. If, for example, ##y = t^2##, then we can write the derivative of y with respect to t as ##\frac{dy}{dt} = 2t## or we can write the differential of y as ##dy = 2t~dt##. The latter form is used all the time in substitutions for integration problems.

    Also, as I mentioned before, separation of variables is a standard technique in virtually all differential equations textbooks, and one that is usually the first technique presented.
     
  7. Feb 28, 2016 #6

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    I was unclear - I was agreeing with you.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Integrating factor
  1. Integrating factors (Replies: 6)

  2. Factoring an Integral (Replies: 8)

  3. Integrating Factor (Replies: 4)

  4. Integrating factor (Replies: 5)

  5. Integrating factor (Replies: 11)

Loading...