*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:
[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.
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 [itex]\int\frac{y'(x)}{y(x)}[/itex] = 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.
Integration is a mathematical process used to find the area under a curve, or the inverse operation of differentiation. It involves finding a function's primitive function, also known as the antiderivative.
The purpose of integrating y''/y is to solve for the function y. This is useful in many applications, such as finding the displacement of an object given its acceleration, or finding the velocity of a moving object given its acceleration function.
To integrate y''/y, you can use the natural logarithm function. The integral can be rewritten as ln(y''), which can then be solved using logarithmic rules. The final solution will be in the form of y = Ce^x, where C is a constant.
Yes, for example, if we have the function y''/y = 2x, we can rewrite it as ln(y'') = 2x, and then solve for y'': y'' = e^(2x). Finally, we integrate again to get y = Ce^(2x), where C is a constant.
Integrating y''/y is commonly used in physics, engineering, and other scientific fields to solve for functions related to motion and change. For example, it can be used to find the position, velocity, and acceleration of an object at any given time, as well as to model the growth of populations or the decay of radioactive substances.