Question

(Sonin-Polya theorem). Let \(p(x)>0\) and \(q(x) \neq 0\) be continuously differentiable on an interval \(I\). If \(p(x) q(x)\) is nonincreasing (nondecreasing) on \(I\), then the absolute values of the relative maxima and minima of every nontrivial solution of the equation $$\frac{d}{d x}\left(p(x) \frac{d y}{d x}\right)+q(x) y=0$$ are nondecreasing (nonincreasing) as \(x\) increases.

Short answer

Question: Show that when \(p(x)q(x)\) is nonincreasing (or nondecreasing) on the interval \(I\), the absolute values of the relative maxima and minima of every nontrivial solution of the differential equation \(\frac{d}{d x}\left(p(x) \frac{d y}{d x}\right)+q(x) y=0\) are nondecreasing (or nonincreasing) as \(x\) increases, respectively. Answer: We first redefine the given differential equation as \(\frac{d^2 y}{dx^2} + \frac{d}{dx}\left(\frac{p'(x)}{p(x)}\right)\frac{d y}{d x} + \frac{q(x)}{p(x)}y = 0\). Analyzing the equation at the relative maxima and minima points, we find that if \(p(x)q(x)\) is nonincreasing on the interval \(I\), the relative extrema of the solution are nondecreasing as \(x\) increases. Conversely, if \(p(x)q(x)\) is nondecreasing, the relative extrema of the solution are nonincreasing as \(x\) increases.

Step-by-step solution

Define the given differential equation

The given differential equation is: $$\frac{d}{d x}\left(p(x) \frac{d y}{d x}\right)+q(x) y=0$$

Analyze properties of differentiable functions

Since \(p(x)\) and \(q(x)\) are continuously differentiable on the interval \(I\), their product, \(p(x)q(x)\), is continuously differentiable on that interval as well. Also, their product is either nonincreasing or nondecreasing on interval \(I\). The conditions of the theorem relate to local extrema of the solutions to the differential equation, so we can try to find when the first derivative of the solution \(y'(x)\) is zero, and then analyze whether those points of relative extrema satisfy the conditions in the theorem.

Prove nondecreasing or nonincreasing nature of the absolute values of relative maxima and minima

Let \(y(x)\) be a nontrivial solution of the given differential equation. At a relative extrema point \(x_0\), we have \(y'(x_0) = 0\). Differentiate the equation with respect to \(x\) to get: $$\frac{d}{dx}\left( \frac{d}{d x}\left(p(x) \frac{d y}{d x}\right)+q(x) y \right) = 0$$ To better understand how \(y'(x)\) behaves, rewrite the equation as follows: $$\frac{d^2 y}{dx^2} + \frac{d}{dx}\left(\frac{p'(x)}{p(x)}\right)\frac{d y}{d x} + \frac{q(x)}{p(x)}y = 0$$ If \(y'(x_0) = 0\), then the equation becomes: $$y''(x_0) + \frac{q(x_0)}{p(x_0)}y(x_0) = 0$$ If \(p(x)q(x)\) is nonincreasing on interval \(I\), then \(p'(x)q(x) + p(x)q'(x) \leq 0\). Evaluate this condition at \(x=x_0\), leading to: $$\left(p'(x_0) - \frac{q(x_0)}{p(x_0)}y(x_0)\right)q(x_0) \leq 0$$ Since \(y''(x_0)=-\frac{q(x_0)}{p(x_0)}y(x_0)\), we can substitute it into the inequality, and we get: $$-y''(x_0)q(x_0) \leq 0$$ Thus, if \(p(x)q(x)\) is nonincreasing on \(I\), then consequently, the product \(-y''(x_0)q(x_0)\) is nonpositive, implying that the relative extrema of the solution, \(y(x)\), are nondecreasing as \(x\) increases. Similarly, if \(p(x)q(x)\) is nondecreasing on interval \(I\), then \(p'(x)q(x) + p(x)q'(x) \geq 0\). We can perform the same analysis as before, leading to: $$-y''(x_0)q(x_0) \geq 0$$ In this case, the relative extrema of the solution, \(y(x)\), are nonincreasing as \(x\) increases. Therefore, we conclude that when \(p(x)q(x)\) is nonincreasing on the interval \(I\), the absolute values of the relative maxima and minima of every nontrivial solution of the given equation are nondecreasing as \(x\) increases. Conversely, if \(p(x)q(x)\) is nondecreasing, then the absolute values of the relative maxima and minima are nonincreasing as \(x\) increases.

Key concepts

Sonin-Polya Theorem

Understanding the Sonin-Polya Theorem is crucial in the study of ordinary differential equations (ODEs). This theorem gives us insight into the behavior of the solutions to certain second-order linear differential equations. It specifically addresses the relationship between the characteristics of continuously differentiable functions and their relative extrema across an interval.

In simple terms, when you have a function that fits the form of the Sonin-Polya condition, you can predict how the relative maximum and minimum values of its solutions change over that interval. If the product of given function parts, in this case, \(p(x)q(x)\), is either always decreasing or increasing, it determines whether the extrema values climb or fall as you move along the \(x\)-axis.

This concept is powerful because it helps identify and analyze the pattern of solution behavior without explicitly solving the differential equation.

continuously differentiable functions

Continuously differentiable functions are functions that have derivatives, which themselves are continuous. This property ensures smoothness in the sense that there are no sharp corners or jumps in the function's graph.

For a context such as in the Sonin-Polya theorem, it means that for functions \(p(x)\) and \(q(x)\), not only do they possess first derivatives, \(p'(x)\) and \(q'(x)\), but those derivatives are continuous across the interval \(I\).

Generally, the conditions for the Sonin-Polya theorem require such continuously differentiable functions because they guarantee predictable behavior over specific intervals, allowing the theorem to apply effectively. Smooth transformations are needed to secure non-abrupt variations which could otherwise affect the expected non-increasing or non-decreasing characteristics of the product of the functions.

relative maxima and minima

Relative maxima and minima refer to points in a function where a local peak or dip occurs. Essentially, these are the points where the function doesn't increase above (maximum) or doesn't decrease below (minimum) in the immediate neighborhood.

Identifying these points on a curve can be done by setting the first derivative \( y'(x) \) of the solution function to zero. This indicates a spot where the function pauses in rising or falling, marking a potential turning point.

The Sonin-Polya theorem uses this concept to determine how these turning points or extremas change as one moves through an interval. Crucially, the nature of these extremas (whether they increase or decrease in magnitude) depends on whether a specific condition holds for the product \(p(x)q(x)\), as mentioned in the solution analysis.

nontrivial solution analysis

Nontrivial solutions of an ordinary differential equation are those solutions which are not constantly zero, implying they contain meaningful and useful information about the system being analyzed. They are essential in understanding the dynamics described by the equation rather than the static or null cases.

In the context of the Sonin-Polya theorem, identifying these nontrivial solutions allows one to study the more active part of a system's response under specific conditions. This is critical because only nontrivial solutions usually present distinct relative maxima and minima necessary for thorough analysis.

With the assumption that the solution is indeed nontrivial, it validates the investigation into the extrema's behavior over an interval – providing insight into how maximal and minimal points evolve, adhering to the Sonin-Polya conditions. Analyzing these further helps in constructing comprehensive solutions for complex differential equations.