(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.