Problem 7 - A First Course in Differential Equations with Modeling Applications

Question

Given that \(x(t)=c_{1} \cos \omega t+c_{2} \sin \omega t\) is the general solution of \(x^{\prime \prime}+\omega^{2} x=0\) on the interval \((-\infty, \infty),\) show that a solution satisfying the initial conditions \(x(0)=x_{0}, x^{\prime}(0)=x_{1}\) is given by \(x(t)=x_{0} \cos \omega t+\frac{x_{1}}{\omega} \sin \omega t\).

Step-by-Step Solution

Answer

Substitute initial conditions into the general solution to find constants.

Step 1: Understanding the problem

We need to show that the function \(x(t)=x_{0} \cos \omega t+\frac{x_{1}}{\omega} \sin \omega t\) satisfies both the differential equation \(x^{\prime \prime}+\omega^{2} x=0\) and the initial conditions \(x(0)=x_{0}, x^{\prime}(0)=x_{1}\).

Step 2: Apply initial conditions

Start with the general solution: \(x(t)=c_{1} \cos \omega t+c_{2} \sin \omega t\). Apply the initial conditions \(x(0)=x_{0}\) and \(x^{\prime}(0)=x_{1}\). This gives the equations \(c_{1} \cos \omega \cdot 0 + c_{2} \sin \omega \cdot 0 = x_{0}\) which simplifies to \(c_{1}=x_{0}\), as \(\cos(0) = 1\) and \(\sin(0) = 0\).

Step 3: Differentiate the general solution

Differentiate \(x(t)=c_{1} \cos \omega t+c_{2} \sin \omega t\) to find \(x'(t)\). This yields \(x'(t) = -c_1 \omega \sin \omega t + c_2 \omega \cos \omega t\). Apply the condition \(x^{\prime}(0)=x_{1}\), which simplifies to \(c_2 \omega = x_1\) because \(\cos(0) = 1\) and \(\sin(0) = 0\). Thus, \(c_2 = \frac{x_1}{\omega}\).

Step 4: Substitute the constants

Substitute \(c_1=x_{0}\) and \(c_2=\frac{x_1}{\omega}\) back into the general solution to get \(x(t)=x_{0} \cos \omega t+\frac{x_{1}}{\omega} \sin \omega t\).

Step 5: Verify the solution

Check if the proposed solution \(x(t)=x_{0} \cos \omega t+\frac{x_{1}}{\omega} \sin \omega t\) satisfies the differential equation. Find the second derivative \(x''(t)\) and substitute into the differential equation \(x^{\prime \prime} + \omega^{2} x=0\). Ensure it simplifies to zero, confirming the solution satisfies the equation.

Key Concepts

General Solution

In the context of differential equations, a general solution represents a family of functions that cover all possible solutions to the equation. Here, we focus on the equation \(x'' + \omega^2 x = 0\). This type of equation is known as a homogeneous second-order linear differential equation.

The general solution for this specific differential equation is given as \(x(t) = c_1 \cos \omega t + c_2 \sin \omega t\). This solution uses the properties of sine and cosine functions, which are periodic in nature to encompass all possible oscillatory motions that the system might possess.

• \(c_1\) and \(c_2\) are constants that will be determined by specific initial conditions.
• \(\omega\) is a constant that often represents angular frequency in physical systems.

Understanding the general solution helps to see how the system's behavior can be described over time without specific starting points.

Initial Conditions

Initial conditions are specific values that determine the exact solution from the general solution of a differential equation. They act like necessary bookmarks telling us which specific path out of the infinite possibilities the solution will actually take.

For our problem, the initial conditions provided are \(x(0) = x_0\) and \(x'(0) = x_1\). By knowing the behavior of the system at time \(t=0\), we can solve for the constants within our general solution.

• The first condition, \(x(0) = x_0\), allows us to find \(c_1\).
• The second condition, \(x'(0) = x_1\), helps us determine \(c_2\).

Applying these conditions ensures that our solution perfectly matches the situation described by the problem at \(t=0\), allowing for a tailored solution for the given scenario.

Harmonic Oscillator

A harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement. The classic example from physics is a mass attached to a spring.

For the differential equation \(x'' + \omega^2 x = 0\), the motion described is characteristic of such an oscillator. The solutions given by \(\cos\) and \(\sin\) functions represent simple harmonic motions.

• The frequency of oscillation is determined by \(\omega\), a constant within the equation.
• The combination of sine and cosine ensures that all possible phase shifts in motion are covered.

Understanding harmonic oscillators is crucial for studying physical systems where wave-like behavior occurs, such as in mechanical vibrations or electrical circuits.

Second Derivative

The second derivative of a function gives us insight into its concavity and the acceleration of the motion it describes. In differential equations, particularly in physics, the second derivative often represents physical acceleration amid a dynamic system.

For the function \(x(t) = c_1 \cos \omega t + c_2 \sin \omega t\), the second derivative, \(x''(t)\), is pivotal. It aids in verifying that our function satisfies the original differential equation, \(x'' + \omega^2 x = 0\).

• Computing \(x''(t)\) results in \(-c_1 \omega^2 \cos \omega t - c_2 \omega^2 \sin \omega t\).
• Adding \(\omega^2 x(t)\) to \(x''(t)\) should equal zero, confirming that the general solution is correct.

Recognizing the role of the second derivative is key in ensuring compliance of a solution with the governing dynamics of a system.