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Differentiating an integral and finding f(x)

  1. Dec 28, 2015 #1
    1. The problem statement, all variables and given/known data
    Find all f(x) satisfying:
    ∫f dx ∫1/f dx = -1

    2. Relevant equations


    3. The attempt at a solution

    I solved for ∫1/f dx and differentiated both sides (using the quotient rule for the right side):
    ∫1/f dx = -1 / ∫f dx
    1/f = f / (∫f dx)2
    (∫f dx)2 = f2
    ∫fdx = ±f

    f = ±f'

    Solving the differential equation for f = f ' I get f = cex

    But when I try to prove if my solution is correct, I got:

    ∫cex dx ∫ 1/(cex) dx = -1
    (ex + k1)(-e-x + k2) = -1
    And I don't know what to do to get -1 on the left side.

    Could you give me a hint please?
     
  2. jcsd
  3. Dec 28, 2015 #2

    SammyS

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    Wow! It's hard to see the ' on ƒ '
    It's been edited. (Sorry for messing up that quote originally.)
    It works if k1 = k2 = 0 .

    Also, where did the constant, c, go in your last line?
     
  4. Dec 28, 2015 #3
    Solving the differential equation for f = f ' I get f = cex

    But when I try to prove if my solution is correct, I got:

    ∫cex dx ∫ 1/(cex) dx = -1
    (ex + k1)(-e-x + k2) = -1
    And I don't know what to do to get -1 on the left side.

    Could you give me a hint please?[/QUOTE]
    It works is k1 = k2 = 0 .

    But is it legal to do that? Giving values to fit the desired result?

    Also, where did the constant, c, go in your last line?[/QUOTE]
    "c" cancelled with the another "c" in the denominator of the second integral.
     
  5. Dec 28, 2015 #4

    SammyS

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    c does not cancel if k1, k2 ≠ 0 .

    You're right.

    I was missing the fact that you can factor c out of the integrals or equivalently, you can have the integration constants "absorb" c.
     
    Last edited: Dec 28, 2015
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