91Ó°ÊÓ

When we talk about the successive zeros of a solution of a differential equation, we refer to the points where the solution curve crosses the x-axis, subsequently returning to zero again after taking some positive or negative values.
To understand this, consider a continuous curve that represents a solution to a differential equation. When it crosses the x-axis at two points, these are its 'successive zeros'. In mathematical terms, if we have a function \( \phi(x) \) with zeros at \( x_1 \) and \( x_2 \), it means \( \phi(x_1) = 0 \) and \( \phi(x_2) = 0 \), with possible non-zero values in between.
If you're working with two different differential equations, as in our original problem, comparing the zeros of their solutions helps us understand the relative behavior of these solutions. It's particularly interesting when these solutions are used under different potential-like functions, \( \alpha(x) \) and \( \beta(x) \), especially if \( \beta(x) > \alpha(x) \). This inequality hints that the solution of the second equation "outgrows" the first one in some sense, morphing the intervals of zeros in a notable way.

The comparison theorem for differential equations is a powerful tool in understanding how solutions to different differential equations behave relative to each other. It gives us insights that are not immediately obvious from solving the differential equations separately.
In the context of the original exercise, the comparison theorem is used to infer the existence of zeros of one solution based on another. Because \( \beta(x) > \alpha(x) \), if \( \phi(x) \) and \( \psi(x) \) are solutions to the equations \( y'' + \alpha(x) y = 0 \) and \( y'' + \beta(x) y = 0 \) respectively, then we can use this theorem to argue that \( \psi(x) \) must also achieve a zero between the successive zeros of \( \phi(x) \).
The theorem essentially ensures that the behavior of the solutions is influenced by the functions \( \alpha(x) \) and \( \beta(x) \) in a predictable way, given the differential equations' constraints. This property is beneficial for analyzing and predicting solutions without explicitly solving them.

The Sturm-Picone theorem is an elegant result in the study of differential equations. It provides a comparison test between two second-order linear differential equations using their solutions' zeros.
According to the theorem, if you have two differential equations \( y'' + p(x) y = 0 \) and \( y'' + q(x) y = 0 \) where \( q(x) > p(x) \), then the zeros of the second equation's solution are more frequent. The theorem gives a rigorous grounding to the idea that the zeros of \( \psi(x) \) will fall between those of \( \phi(x) \).
The application of the Sturm-Picone theorem involves calculating or estimating where and how the solutions cross the x-axis. This gives us a way to understand how solutions "bounce" between each other's zero intervals. The deeper implications of the theorem lie in its ability to predict oscillatory behavior of the solutions, offering a window into the dynamics of the systems modeled by such differential equations.