Question

(a) For what nonzero values of \(\mathrm{k}\) does the function \(\mathrm{y}=\sin \mathrm{kt}\) satisfy the differential equation \(y^{\prime \prime}+9 y=0 ?\) (b) For those values of \(k\), verify that every member of the family of functions \(\mathrm{y}=\mathrm{A} \sin \mathrm{kt}\) + b cos kt is also a solution.

Short answer

Question: Find the nonzero values of k such that the function \(y = \sin(kt)\) satisfies the second-order differential equation \(y^{\prime \prime} + 9y = 0\), and verify that every member of the family of functions \(y = A\sin(kt) + B\cos(kt)\) is also a solution for those values of k. Answer: The nonzero values of k that satisfy the given differential equation are \(k = 3\) and \(k = -3\). The family of functions \(y = A\sin(kt) + B\cos(kt)\) is a solution for these values of k as well.

Step-by-step solution

Find the derivatives

Find the first and second derivatives of y with respect to t. First derivative: \(y' = \frac{dy}{dt} = k\cos(kt)\) Second derivative: \(y'' = \frac{d^2y}{dt^2} = -k^2\sin(kt)\)

Substitute in the differential equation

Substitute the second derivative (\(y''\)) and original function (\(y\)) into the given differential equation \(y'' + 9y = 0\) and then solve for k. \(-k^2\sin(kt) + 9\sin(kt) = 0\) Factor out \(\sin(kt)\) in the equation: \(\sin(kt)(-k^2 + 9) = 0\)

Solve for k

Solve the equation for nonzero values of k. Since \(\sin(kt)\) is nonzero, we are left with: \(-k^2 + 9 = 0\) \(k^2 = 9\) \(k = \pm 3\) So, the nonzero values of k that satisfy the given differential equation are \(k = 3\) and \(k = -3\).

Verify the family of functions

Verify that the family of functions given by \(y = A\sin(kt) + B\cos(kt)\) is a solution for \(k = 3\) and \(k = -3\). Here, we will use the Chain Rule for differentiation. First, find the first and second derivatives for the family of functions: First derivative: \(y' = 3A\cos(kt) - 3B\sin(kt)\) Second derivative: \(y'' = -9A\sin(kt) - 9B\cos(kt)\) Now, substitute these derivatives and the family of functions into the given differential equation: \(-9A\sin(kt) - 9B\cos(kt) + 9(A\sin(kt) + B\cos(kt)) = 0\) Simplify the equation: \(0\sin(kt) + 0\cos(kt) = 0\) This equation is true for all values of t, A, and B. Thus, the family of functions \(y = A\sin(kt) + B\cos(kt)\) is a solution to the given differential equation for the values of \(k = 3\) and \(k = -3\).

Key concepts

Solving Differential Equations

Understanding differential equations is fundamental for a variety of disciplines such as physics, engineering, and mathematics. Essentially, these equations describe relationships involving rates of change and the values of quantities themselves.

At the core of solving a differential equation is finding a function or a set of functions that satisfy the equation. The process often involves

• Identifying the type of differential equation (e.g., linear, separable, homogeneous)
• Finding the general solution using techniques like separation of variables or integrating factors
• Applying initial or boundary conditions to find the particular solution when required.

Let's remember, the ultimate goal is to find an unknown function that, when substituted into the differential equation, makes the equation true.

Second Order Differential Equations

A second order differential equation includes, at least, a second derivative of an unknown function. These equations are particularly important because they appear frequently in the modeling of natural systems, including oscillations and waves.

To solve these, one might:

• Use characteristic equations for linear, constant coefficient second-order DEs
• Employ methods such as undetermined coefficients or variation of parameters for nonhomogeneous equations
• Consider the application of boundary or initial conditions to ensure solutions are relevant to the physical context.

In the exercise, we are keen on finding the specific values of the parameter k that lead to meaningful solutions.

Harmonic Motion Equations

Harmonic motion is epitomized by the sinusoidal movement found in systems like springs and pendulums. An understanding of harmonic motion equations is indispensable in both classical and modern physics.

The canonical form of the simple harmonic motion equation is a second order differential equation, typically written as \( y'' + \omega^2y = 0 \), where \( \omega \) is the angular frequency of the motion. This reflects the restoring force being directly proportional to the displacement but in the opposite direction.

Harmonic motion illustrates fundamental concepts such as resonant frequencies and normal modes, which are not only critical in physics but also in engineering domains like mechanical system design and signal processing.

Sinusoidal Function Solutions

Sinusoidal functions, which encompass both sine and cosine functions, frequently emerge as solutions to second order differential equations involving harmonic motion. Their periodic nature represents oscillations and waves accurately.

In the context of our exercise, when the differential equation \( y'' + 9y = 0 \) was explored, the solution involved sinusoidal functions with specific values for k. This is expected since sinusoidal functions are characteristic solutions for such equations, demonstrating behaviors such as frequency and amplitude which are physically interpretable in the context of waves or oscillations.

The ability to manipulate and understand these sinusoidal function solutions is crucial when examining the dynamics of systems undergoing harmonic motion, as it allows for the exploration of important concepts like damping, forced oscillations, and resonance.