Question

Think About It It is known that \(y=A \sin \omega t\) is a solution of the differential equation \(y^{\prime \prime}+16 y=0 .\) Find the values of \(\omega .\)

Short answer

The values for \( \omega \) are \( \pm 4 \).

Step-by-step solution

Differentiate the function once

The first derivative of the given function \(y = A \sin(\omega t)\) with respect to \(t\) is \(y' = A \omega \cos(\omega t)\). This represents the velocity of the motion at any time \(t\).

Differentiate the function again

The second derivative of the given function with respect to \(t\) is \(y'' = -A \omega^2 \sin(\omega t)\). This represents the acceleration of the motion at any time \(t\).

Substitute into the given differential equation

Now substitute \(y''\) into the given differential equation \(y'' + 16y = 0\). Substituting gives us \(-A \omega^2 \sin(\omega t) + 16A \sin(\omega t) = 0\). Simplifying yields to \(- \omega^2 + 16 = 0\).

Solve for \( \omega \)

To find the values of \( \omega \), solve the equation \(- \omega^2 + 16 = 0\), which results in \( \omega^2 = 16\), thus \( \omega = \pm 4\).

Key concepts

Differential Equation Solutions

Understanding differential equations is crucial for analyzing phenomena in fields like physics and engineering. A differential equation is a mathematical equation that relates some function with its derivatives. Solving a differential equation means finding the function that satisfies the equation.

In our example, the differential equation is second-order, meaning it involves the second derivative of the function. The most common methods for solving these equations include the characteristic equation method, undetermined coefficients, or variation of parameters. It's also worth noting that differential equations can have more than one solution, often referred to as the general solution, which includes arbitrary constants that can be determined if initial conditions are known.

Sinusoidal Functions

Sinusoidal functions, such as sine and cosine, are periodic and represent oscillating motions. These functions are fundamental in modeling situations that involve repetitive patterns, like the motion of a pendulum or alternating electrical currents.

The general form of a sinusoidal function is defined as either \( A \text{sin}(\omega t + \phi) \) or \( A \text{cos}(\omega t + \phi) \) where \( A \) represents the amplitude, \( \omega \) denotes the angular frequency, \( t \) is time, and \( \phi \) is the phase shift. Understanding the properties of sinusoidal functions is key to solving differential equations that describe wave-like or harmonic motions.

Derivatives

In calculus, derivatives represent the rate at which a function is changing at any given point and are foundational to dynamic systems analysis.

The first derivative of a function gives us the velocity of an object if the function represents its position. The second derivative provides the acceleration. In our exercise, by performing the derivative of the sinusoidal function \( y = A \text{sin}(\omega t) \) twice, we identified the relationship between the function and its second derivative, which is an essential step in solving the given differential equation. Calculating derivatives is, therefore, a pivotal skill in both theoretical and applied science.

Harmonic Motion

Harmonic motion refers to a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. One classic example of harmonic motion is a mass attached to a spring.

In the context of our exercise, the differential equation \( y'' + 16y = 0 \) describes a simple harmonic oscillator with no damping or external forces. The solution involves sinusoidal functions because they naturally encapsulate the properties of harmonic motion—oscillation and periodicity. Through solving the differential equation, we can understand the behavior of the system and predict its motion over time.