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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.
     
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