Question

Verify by direct substitution that the function \(\phi(t)=A \sin (\omega t)+B \cos (\omega t)\) of (1.56) is a solution of the second-order differential equation \((1.55), \ddot{\phi}=-\omega^{2} \phi .\) (since this solution involves two arbitrary constants - the coefficients of the sine and cosine functions \(-\) it is in fact the general solution.)

Short answer

The function \(\phi(t)\) satisfies the differential equation by substitution.

Step-by-step solution

Differentiate the given function \( \phi(t) \)

First, find the first derivative of \( \phi(t) = A \sin(\omega t) + B \cos(\omega t) \) with respect to \( t \). Using the standard derivative rules, the derivative is \( \phi'(t) = A \omega \cos(\omega t) - B \omega \sin(\omega t) \).

Differentiate again to find \( \ddot{\phi} \)

Differentiate \( \phi'(t) = A \omega \cos(\omega t) - B \omega \sin(\omega t) \) to find the second derivative. Utilizing the standard derivatives again, the result is \( \ddot{\phi}(t) = -A \omega^2 \sin(\omega t) - B \omega^2 \cos(\omega t) \).

Substitute into the differential equation

Substitute \( \ddot{\phi}(t) = -A \omega^2 \sin(\omega t) - B \omega^2 \cos(\omega t) \) into the equation \( \ddot{\phi} = -\omega^2 \phi \). Check if both sides are equal: \(-A \omega^2 \sin(\omega t) - B \omega^2 \cos(\omega t) = -\omega^2 [A \sin(\omega t) + B \cos(\omega t)]\), which simplifies to the same expression as the left side.

Key concepts

General Solution

A general solution to a differential equation is a formula that encompasses all possible solutions of that equation. This means it contains arbitrary constants that can be adjusted to fit specific boundary conditions or initial values. In our problem, the general solution involves the equation \[ \phi(t) = A \sin(\omega t) + B \cos(\omega t) \] Here, \( A \) and \( B \) are the arbitrary constants. These are adjustable and allow the solution to cater to any initial conditions presented by a problem. The function \( \phi(t) \) being a solution means every combination of \( A \) and \( B \) satisfies the equation \( \ddot{\phi} = -\omega^2 \phi \). Therefore, it provides a comprehensive set of solutions for different scenarios. Having such a general form is useful because it provides flexibility in solving differential equations related to waves, oscillations, and other periodic phenomena.

Sine and Cosine Functions

Sine and cosine functions are fundamental in the study of waves and oscillations. In the equation \( \phi(t) = A \sin(\omega t) + B \cos(\omega t) \), both functions contribute to the behavior of the solution. These trigonometric functions are periodic and continuous, characteristics that make them ideal for describing oscillatory systems like springs and pendulums.

Key characteristics include:

• Periodicity: Both sine and cosine repeat their values in cyclic patterns, having a period of \( 2\pi \).
• Amplitude Modulation: The constants \( A \) and \( B \) determine the amplitude of each function, reflecting the maximum extent of oscillation.
• Phase Relationship: Sine and cosine functions are phase-shifted by \( \pi/2 \) radians; this is crucial in modeling waves where such phase shifts create interference patterns.

These functions have wide applications, not only in simple harmonic motion but also in Fourier analysis, which breaks down more complex waves into sums of sine and cosine components.

Differentiation

Differentiation is a process of finding the rate at which a function is changing. In the context of differential equations, differentiation helps us understand how quantities evolve over time. For the function \( \phi(t) = A \sin(\omega t) + B \cos(\omega t) \), differentiation involves:

• First Derivative: Differentiating \( \phi(t) \) with respect to \( t \) gives \( \phi'(t) = A \omega \cos(\omega t) - B \omega \sin(\omega t) \). This represents the instantaneous rate of change of the function.
• Second Derivative: Further differentiation gives \( \ddot{\phi}(t) = -A \omega^2 \sin(\omega t) - B \omega^2 \cos(\omega t) \). The second derivative is crucial as it reveals how the rate of change itself changes, often related to acceleration and force in physics.

Differentiation is indispensable in solving differential equations since it allows us to work backwards from a general solution and verify if it satisfies the given conditions.

Direct Substitution

Direct substitution is a technique used to verify if a particular solution satisfies a given equation. It involves plugging the solution and its derivatives back into the equation to ensure consistency and correctness. In our case:

• Start by differentiating \( \phi(t) \) to find both \( \phi'(t) \) and \( \ddot{\phi}(t) \).
• Substitute \( \ddot{\phi}(t) = -A \omega^2 \sin(\omega t) - B \omega^2 \cos(\omega t) \) into the differential equation \( \ddot{\phi} = -\omega^2 \phi \).
• Ensure that this substitution leads to equivalent expressions on both sides of the equation.

Successful substitution implies that \( \phi(t) \) is indeed a valid solution to the differential equation. This straightforward process confirms the relationship between the solution and the equation, providing confidence in the derived general solution.