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Integrating y''/y

  1. Aug 29, 2011 #1
    1. The problem statement, all variables and given/known data

    How would you find the integral of y''/y (with respect to x)?

    2. Relevant equations



    3. The attempt at a solution

    I have absolutely no idea how to begin. All I know is that if it were y'/y, it would be log(y) + c. Perhaps the integral here has something to do with logs.

    Thanks a lot.
     
  2. jcsd
  3. Aug 29, 2011 #2
    Try integration in parts, where f(x) = y''(x), g'(x) = 1/y(x).
    You should arrive to a new integral you can solve...
     
  4. Aug 30, 2011 #3

    HallsofIvy

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    Tomer's suggestion is good but I think a more common notation would be u= 1/y, dv= y'' dy.
     
  5. Aug 30, 2011 #4

    LCKurtz

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    I don't see how either of these substitutions help to solve this problem. y is a function of x and the variable of integration is x.
     
  6. Aug 30, 2011 #5
    I haven't for a second assumed x isn't the variable of integration.
    However, trying to solve it now on paper I noticed that what I first though would be a solvable integral isn't really.
    Do you have any ideas then?
     
  7. Aug 30, 2011 #6
    If:

    [tex]\frac{d}{dx}\left[\frac{y'}{y}+\left(\frac{d}{dx}\log y\right)^2\right]=\frac{y''}{y}[/tex]

    then isn't:

    [tex]\int \frac{y''}{y}=\frac{y'}{y}+\left(\frac{d}{dx}\log y\right)^2[/tex]

    Not entirely sure guys. Just a start.
     
  8. Aug 30, 2011 #7
    What you say is right, but I don't see how the first formula you wrote is correct.
     
  9. Aug 30, 2011 #8
    Ok, it's wrong. Sorry.
     
  10. Aug 30, 2011 #9
    It's ok, apparently we're all wrong :-)
     
  11. Aug 30, 2011 #10

    PAllen

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    No, it seems that the starting point in post #6 can't possibly be right. It could only be right if the derivative of the second term in brackets is zero, which can't possibly be true, in general.
     
  12. Aug 30, 2011 #11

    LCKurtz

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    I will be a bit more emphatic. You aren't going to find a nice closed form general solution with any such techniques.
     
  13. Aug 30, 2011 #12

    PAllen

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    Yeah, it is equivalent to asking what is a general formula for the solution of the following homogeneous, linear second order equation with non-constant coefficients:

    y'' - f(x) y = 0

    which is absurd (unless f(x) is special in some way).
     
  14. Aug 30, 2011 #13
    Why is this absurd? If I can say that [itex]\int\frac{y'(x)}{y(x)}[/itex] = ln(y(x)), which is a closed form general solution, why should the given integral be absurd?

    I don't see any way to find a closed formula, but I also don't see why the question should be meaningless.
     
    Last edited: Aug 30, 2011
  15. Aug 30, 2011 #14

    PAllen

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    It isn't meaningless, it is just well understood - it has been studied (stated as diff.eq.) for centuries. All facts about it are known. It is known that there are solutions for many common f(x), but no formula for a general solution.
     
  16. Aug 31, 2011 #15
    I understand :p
     
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