Variational methods conjugate points

[braindead101]

(a will be alpha and b will be beta)
Let y=y(x,a,b) be a general solution of Euler's equation, depending on two parameters a and b. Prove that if the ratio (subdifferential y/subdifferential a)/(subdifferential y/subdifferential b) is the same at the points, the points are conjugate.

I cannot find the theorem to use for this. i am suppose to take the derivative with respect to a and b and find the ratio. but i am looking for that something to take the derivative of.

As well, the definition of conjugate points is as follow:
The point a tilda (does not equal a) is said to be conjugate to the point a if the equation -d/dx(Ph')+Qh = 0 has a solution which vanishes for x=a and x=a tilda, but is not identically 0.

I have been looking over and over on the internet for this, I cannot show any work as I really do not know how to do it without the proper theorem.

 

[lippyka]

Any help would be appreciated. To prove that the points are conjugate, you need to show that the equation -d/dx(Ph')+Qh = 0 has a solution which vanishes for both x=a and x=a tilda (where a tilda is not equal to a). You can do this by taking derivatives of y with respect to a and b and finding the ratio between them as you mentioned. Let's denote the derivatives of y with respect to a and b as y'a and y'b respectively. Then the ratio between them is given by: R = (y'a/y'b)If the ratio R is the same for both points x=a and x=a tilda, then this suggests that the equation -d/dx(Ph')+Qh = 0 has a solution which vanishes at both points. Therefore these points are conjugate.