Verify that \(y=A \sin k x+B \cos k x\), where \(A, B\), \(k\) constants is a solution of $$ y^{\prime \prime}+k^{2} y=0 $$

Understanding the second derivative is crucial when studying differential equations. It represents the rate at which the first derivative is changing, providing information about the curvature or 'concavity' of the graph of a function. In simpler terms, while the first derivative tells us about the slope of the function, the second derivative tells us how that slope is changing.

In the context of harmonic motion, the second derivative is particularly significant as it relates to acceleration. When we analyze the motion of a particle that is undergoing simple harmonic motion, the second derivative of its position with respect to time gives us the acceleration of the particle. For instance, if the position of a particle is described by the function \(y(t)\), then the acceleration of the particle is described by the function \(y''(t)\).

In the given differential equation \(y'' + k^2y = 0\), finding the second derivative of the proposed solution helps us to determine whether the function is indeed a solution to the equation. It shows that the acceleration (second derivative) of the function is directly proportional to the function itself, a characteristic feature of harmonic motion.

Trigonometric functions such as sine and cosine are periodic functions that have a plethora of applications in different fields, including physics, engineering, and mathematics. These functions are fundamental in describing oscillatory motions, like waves and vibrations.

The trigonometric functions \(\text{sin}(x)\) and \(\text{cos}(x)\) have a special relationship with second derivatives. When differentiated twice, each function cycles back to itself, scaled by a constant. This property makes these functions natural solutions to second-order differential equations describing harmonic motion. The general solution provided for the equation, \(y = A\sin{kx} + B\cos{kx}\), showcases a linear combination of sine and cosine functions, which is common in the solutions of equations governing oscillatory behavior.

Harmonic motion is a type of periodic movement where the restoring force is directly proportional to the displacement and acts in the opposite direction. This is best exemplified by objects like springs and pendulums.Simple harmonic motion can be described using sinusoidal functions because these functions inherently exhibit the properties of oscillation. The differential equation \(y'' + k^2y = 0\) describes a system in harmonic motion, where \(k^2\) is a constant related to the properties of the system, such as mass and spring stiffness in mechanical systems, or inductance and capacitance in electrical systems.The solution \(y = A\sin{kx} + B\cos{kx}\) combines both sine and cosine functions, representing the displacement of the system in motion over time. The constants \(A\) and \(B\) may represent initial conditions such as the initial displacement and velocity.

The chain rule is a basic but powerful tool in calculus that is used to differentiate composite functions. When you have a function inside another function, the chain rule allows you to find the derivative by multiplying the derivative of the outer function by the derivative of the inner function.

For example, with a function like \(f(g(x))\), the derivative with respect to \(x\) is \(f'(g(x))g'(x)\). In the context of the exercise, the chain rule is utilized to differentiate trigonometric functions where the argument is itself a function of \(x\), such as \(\text{sin}(kx)\) and \(\text{cos}(kx)\). The differentiation of these functions with respect to \(x\) leads to the terms \(Ak\text{cos}(kx)\) and \(Bk\text{sin}(kx)\), respectively, with \(k\) being part of the derivative due to the chain rule application.