Question

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

Short answer

Using the Sturm-Picone Comparison Theorem, demonstrate that any nontrivial solution of the given second-order differential equation $$y^{\prime \prime}+(\sinh x) y=0$$ has at most one zero in \((-\infty, 0)\) and infinitely many zeros in \((0, \infty)\).

Step-by-step solution

Choose an auxiliary equation

We choose an auxiliary equation for which the Sturm-Picone Comparison Theorem can be applied directly. In this case, let's consider the equation: $$y^{\prime \prime} + y = 0$$ which has a simpler harmonic oscillator form.

Analyze the auxiliary equation

The auxiliary equation is a linear homogeneous second-order differential equation with constant coefficients. We can find the solutions to this equation using the characteristic equation method. The characteristic equation will be: $$r^2 + 1 = 0$$ This equation has roots \(r = \pm i\). Therefore, the general solution of the auxiliary equation is: $$y(x) = c_1 \cos x + c_2 \sin x$$ where \(c_1\) and \(c_2\) are arbitrary constants.

Apply Sturm-Picone Comparison Theorem

According to the Sturm-Picone Comparison Theorem, we can compare the number of zeros of nontrivial solutions to our given differential equation with the zeros of nontrivial solutions of the auxiliary equation. We analyze the zeros of the auxiliary equation's general solution. Since \(c_1\cos x + c_2\sin x\) is sinusoidal, there will be infinitely many zeros in \((0, \infty)\) assuming \(c_1\) and \(c_2\) are non-zero simultaneously. Also, for this general solution, we can have at most one zero in \((-\infty, 0)\).

Conclusion

From the Sturm-Picone Comparison Theorem, it can be concluded that if a nontrivial solution of our given differential equation $$y^{\prime \prime}+(\sinh x) y=0$$ exists, it will have at most one zero in \((-\infty, 0)\) and infinitely many zeros in \((0, \infty)\).

Key concepts

Sturm-Picone Comparison Theorem

The Sturm-Picone Comparison Theorem is a powerful mathematical tool for analyzing the behavior of solutions to differential equations. Primarily, it helps compare the zeros of two different second-order linear differential equations. This theorem is especially useful when dealing with complex equations, as it allows you to infer zero-related properties of one equation based on a simpler auxiliary equation.

In our exercise, we utilized the Sturm-Picone Comparison Theorem to compare the given differential equation with a simpler one: the harmonic oscillator equation. Here's how it works in essence:

• First, you identify or choose an auxiliary equation that resembles the behavior or structure of the original equation but is easier to solve.
• Next, solve the auxiliary equation to understand its solutions and, particularly, the distribution of its zeros.
• Finally, apply the Sturm-Picone Comparison Theorem to draw conclusions about the zeros of the original equation.

This theorem identifies the potential zero-finding not directly by solving the complex equation but by drawing parallels with another, often more accessible equation. This comparison confirmed that there is at most one zero in he negative region or the given differential equation and infinitely many in the positive range.

Auxiliary Equation

An auxiliary equation is a secondary equation used to facilitate solving or understanding a more complex primary equation. In the context of differential equations, an auxiliary equation often has a simpler form, making computations more manageable.

In our problem, we considered the auxiliary equation:
\[ y'' + y = 0 \]
This particular choice is due to its simplicity, resembling the behavior of harmonic oscillators, and it’s often selected because it can be readily solved.

• The characteristic equation for this auxiliary differential equation is \( r^2 + 1 = 0 \), deriving from assuming solutions of the form \( y = e^{rx} \).
• The roots of this characteristic equation are \( r = \pm i \), indicating oscillatory solutions.
• Thus, the general solution takes the form \( y(x) = c_1 \cos x + c_2 \sin x \), where \( c_1 \) and \( c_2 \) are constants that adjust the amplitude and phase

By solving the auxiliary equation, we gain insights into the behavior of its solutions, particularly their zero-crossings, aiding in the comparison with the primary equation.

Linear Homogeneous Second-Order Differential Equation

A linear homogeneous second-order differential equation is a differential equation of the form:
\[ a(x) y'' + b(x) y' + c(x) y = 0 \]
where \( a(x)\), \( b(x) \), and \( c(x) \) are functions of the variable \( x \). The term "linear" implies that each term involving \(y\) and its derivatives is linear, and "homogeneous" indicates that the equation equals zero.

This type of equation frequently appears in various scientific and engineering disciplines. In this context, our focus differential equation is linear and homogeneous
\[ y'' + \sinh(x) y = 0 \]
The primary goal is often to find solutions for \( y \) where it satisfies the differential equation.

• A second-order differential equation implies that the highest derivative involved is the second derivative, \( y'' \).
• "Linear" means no products or powers of \( y \), \( y' \), or \( y'' \) beyond the first power and no higher order terms.
• "Homogeneous" specifies the whole expression sums or equals zero rather than a non-zero function.

Understanding these terms helps in classifying the problem and applying appropriate methods for finding the solutions and analyzing their properties.