Question

Show that every nontrivial solution of the equation $$y^{\prime \prime}+e^{x} y=0$$ has infinitely many zeros on \((0, \infty)\), whereas a nontrivial solution of the equation $$y^{\prime \prime}-e^{x} y=0$$ has at most one zero on the interval \((0, \infty)\).

Short answer

Answer: For the equation \(y^{\prime\prime} + e^x y = 0\), a nontrivial solution within the interval (0, ∞) has infinitely many zeros. However, for the equation \(y^{\prime\prime} - e^x y = 0\), a nontrivial solution within the interval (0, ∞) has at most one zero.

Step-by-step solution

Part 1: Infinitely many zeros for \(y^{\prime\prime} + e^x y = 0\)

To analyze this equation, let's consider a nontrivial solution \(y(x)\) with a zero at \(x=x_1\), i.e., \(y(x_1) = 0\). Then, we can analyze the behavior of the solution around this zero point. If \(y'(x_1)\) is positive, then by the equation governing the solution, \(y^{\prime\prime}(x_1) = -e^{x_1} y(x_1) = 0\). Therefore, the solution increases with zero acceleration at this point. On the other hand, if \(y'(x_1)\) is negative, then \(y^{\prime\prime}(x_1)\) is positive by the same argument as before, so the solution decreases but has an upward concavity. In both cases, we can see that the solution must cross the \(x\)-axis again, creating another zero. By continuing this argument, we can deduce that there must be infinitely many zeros in any nontrivial solution of this equation.

Part 2: At most one zero for \(y^{\prime\prime} - e^x y = 0\)

Now, let's consider the second equation, \(y^{\prime\prime} - e^x y = 0\). Again, assume a nontrivial solution \(y(x)\) with a zero at \(x=x_2\), i.e., \(y(x_2) = 0\). Now, we analyze the behavior of the solution around this zero. If \(y'(x_2)\) is positive, then, by the equation, \(y^{\prime\prime}(x_2) = e^{x_2} y(x_2) = 0\). Therefore, the solution increases with zero acceleration. However, when \(y'(x_2)\) is negative, then \(y^{\prime\prime}(x_2)\) is still zero by the same argument. This situation implies that the solution will decrease but has no reason to change its concavity and cross the \(x\)-axis again. In both cases, we can see that any nontrivial solution of this equation will have at most one zero on the interval \((0, \infty)\). So we have proven that a nontrivial solution of the equation \(y^{\prime\prime} + e^x y = 0\) has infinitely many zeros on the interval \((0, \infty)\), whereas a nontrivial solution of the equation \(y^{\prime\prime} - e^x y = 0\) has at most one zero on the same interval.

Key concepts

Nontrivial Solutions

In the context of differential equations, a nontrivial solution refers to any solution that is not identically zero. This means that the function representing the solution has some nonzero value over its domain. Nontrivial solutions are significant because they reflect more complex behaviors of a system governed by a differential equation.

For the equation \(y^{\prime \prime} + e^{x} y=0\), any potential nontrivial solution indicates that there are factors causing the solution to deviate from flatness or triviality. This is crucial because the presence of nontrivial solutions can often lead to interesting dynamics, including the occurrence of zeros (points where the solution crosses the x-axis).

In contrast, the equation \(y^{\prime \prime} - e^{x} y=0\) also allows for nontrivial solutions, but they exhibit different characteristics, specifically concerning their zeros. This variance is related to the nature of the coefficient of \(y\) in the equations, affecting the solutions' behavior and their zeros.

Zeros of Differential Equations

The zeros of differential equations are the values of \(x\) at which the solution \(y(x)\) equals zero. These points are significant in analyzing the behavior of solutions over a domain, particularly in how solutions oscillate.

For the equation \(y^{\prime \prime} + e^{x} y = 0\), it was shown that nontrivial solutions have infinitely many zeros. This occurs because the solutions can oscillate repeatedly across the x-axis, never settling down to being zero for an infinite interval. It's like a pendulum that keeps swinging forever without stopping.

Conversely, the equation \(y^{\prime \prime} - e^{x} y = 0\) has nontrivial solutions that exhibit at most one zero. The solutions can start off crossing the x-axis but will eventually diverge or tend towards one direction without returning to cross again, indicating a single possible point of zero.

Understanding zeros helps in comprehending the complete behavior of differential equations, determining stability, and predicting long-term behavior.

Second-Order Differential Equations

Second-order differential equations involve the second derivative of a function, and they describe a wide range of physical phenomena, from mechanical vibrations to electrical circuits. They are often expressed in the form \(y^{\prime \prime} + P(x) y^{\prime} + Q(x) y = 0\), where \(P(x)\) and \(Q(x)\) are functions that provide additional structure or constraints to the system.

In our examples, both equations belong to the category of second-order linear homogeneous differential equations. Such equations can describe systems with repeated oscillations or decaying amplitudes based on their characteristic equations.

The study of second-order differential equations involves understanding their solutions, which may have complex variables like harmonics or exponential growth/decay, depending on the nature of \(P(x)\) and \(Q(x)\). Specifically, solutions to these equations can often be characterized in terms of their zeros, periodicity, and limits, drawing a link between theory and physical phenomena.

This understanding is crucial for predicting and manipulating real-world applications modeled by these equations.