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: ODE's, Find all functions f that help satisfy the equation

  1. Sep 29, 2012 #1
    1. The problem statement, all variables and given/known data
    Find all functions f that help satisfy the equation
    [itex]\left(\int f(x)dx\right)\left(\int\frac{1}{f(x)}dx\right) = -1[/itex]

    2. Relevant equations

    3. The attempt at a solution
    I'm not quite sure what to do here. When I differentiate I get [itex]f(x)\int1/ f(x) dx + 1/f(x)\int f(x) dx[/itex] which doesn't seem to help if I keep differentiating. I'm kinda stuck on this problem so any help is appreciated. Thanks
  2. jcsd
  3. Sep 30, 2012 #2
    I've been looking at this and trying to figure it out.

    At first, I was thinking of a trig function, but after more thinking that seems to be out. Now I'm thinking something like [itex]f(x) = e^x[/itex], though I haven't really tried to work it out past that part, just an idea for you.
  4. Sep 30, 2012 #3
    The answer clearly seems to be f(x) = e**x I just can't get to proving it.
  5. Sep 30, 2012 #4
    That's the main trouble I was having. I get how to prove that it's not other functions by solving [itex]\int x^n * \int \frac{1}{x^n}[/itex], which shows that a standard function to degree n will always result in something such as [itex]\frac{-x^{n+1}}{n+1}[/itex], if [itex]|n|>1[/itex]. Knowing this shows that you must have a function that repeats itself when integrated, such as a trig function or [itex]e^x[/itex]. This is the closest thing that I've gotten to so far, but it still isn't much of a mathematical proof at all.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook