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

  • Thread starter MeMoses
  • Start date
  • #1
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Homework Statement


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]


Homework Equations





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
 

Answers and Replies

  • #2
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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.
 
  • #3
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The answer clearly seems to be f(x) = e**x I just can't get to proving it.
 
  • #4
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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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