The discussion revolves around finding the integral of the expression y''/y with respect to x. Participants are exploring the implications of this integral within the context of differential equations and logarithmic relationships.
The discussion is ongoing, with various approaches being proposed and questioned. Some participants express skepticism about the possibility of finding a closed-form solution, while others argue that the question remains meaningful despite the challenges. There is a recognition that the integral relates to well-studied differential equations.
Participants are grappling with the complexity of the integral and the assumptions about the function y and its derivatives. There is a shared understanding that the integral may not yield a straightforward solution, particularly due to the non-constant nature of the coefficients involved.
*FaerieLight* said:Homework Statement
How would you find the integral of y''/y (with respect to x)?
Homework Equations
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.
Tomer said:Try integration in parts, where f(x) = y''(x), g'(x) = 1/y(x).
You should arrive to a new integral you can solve...
HallsofIvy said:Tomer's suggestion is good but I think a more common notation would be u= 1/y, dv= y'' dy.
LCKurtz said: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.
jackmell said:If:
\frac{d}{dx}\left[\frac{y'}{y}+\left(\frac{d}{dx}\log y\right)^2\right]=\frac{y''}{y}
then isn't:
\int \frac{y''}{y}=\frac{y'}{y}+\left(\frac{d}{dx}\log y\right)^2
Not entirely sure guys. Just a start.
Tomer said:What you say is right, but I don't see how the first formula you wrote is correct.
Tomer said:It's ok, apparently we're all wrong :-)
LCKurtz said:I will be a bit more emphatic. You aren't going to find a nice closed form general solution with any such techniques.
PAllen said: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).
Tomer said:Why is this absurd? If I can say that \int\frac{y'(x)}{y(x)} = ln(y(x)), which is a closed form general solution (and is equivalent to finding f(x) in the equation y'(x) +f(x)y(x) = 0), 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.
PAllen said: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.