Problem 7 - Advanced Engineering Mathematics

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

The specific solution is \(x(t) = x_0 \cos \omega t + \frac{x_1}{\omega} \sin \omega t\).

Step 1: Determine the general solution form

Given that \(x(t) = c_1 \cos \omega t + c_2 \sin \omega t\) is the general solution of the differential equation \(x'' + \omega^2 x = 0\), the given function already satisfies the equation for any constants \(c_1\) and \(c_2\).

Step 2: Apply initial conditions

We need the function \(x(t)\) to satisfy the initial conditions \(x(0) = x_0\) and \(x'(0) = x_1\). Substitute \(t = 0\) into the general solution to find that:\[x(0) = c_1 \cos(\omega \cdot 0) + c_2 \sin(\omega \cdot 0) = c_1 = x_0.\]Thus, \(c_1 = x_0\).

Step 3: Differentiate the general solution

Differentiate \(x(t) = c_1 \cos \omega t + c_2 \sin \omega t\) with respect to \(t\):\[x'(t) = -c_1 \omega \sin \omega t + c_2 \omega \cos \omega t.\]

Step 4: Apply the derivative initial condition

Substitute \(t = 0\) into the differentiated equation:\[x'(0) = -c_1 \omega \sin(\omega \cdot 0) + c_2 \omega \cos(\omega \cdot 0) = c_2 \omega = x_1.\]Thus, \(c_2 = \frac{x_1}{\omega}\).

Step 5: Substitute constants into the general solution

Substitute \(c_1 = x_0\) and \(c_2 = \frac{x_1}{\omega}\) back into the general solution:\[x(t) = x_0 \cos \omega t + \frac{x_1}{\omega} \sin \omega t.\]

Step 6: Verify the solution

Check that the expression \(x(t) = x_0 \cos \omega t + \frac{x_1}{\omega} \sin \omega t\) satisfies both initial conditions:- At \(t = 0\), \(x(t) = x_0 \cos(0) + \frac{x_1}{\omega} \sin(0) = x_0\).- At \(t = 0\), \(x'(t) = -x_0 \omega \sin(0) + \frac{x_1}{\omega} \omega \cos(0) = x_1\).Thus, the initial conditions are satisfied.

Key Concepts

Harmonic Oscillators

A harmonic oscillator is a system in physics that experiences a restoring force proportional to the displacement from its equilibrium position. This is often encountered in springs and pendulums. The mathematical model of a harmonic oscillator can be represented by a second-order differential equation such as \[ x'' + \omega^2 x = 0.\] In this equation, \(x(t)\) refers to the position as a function of time, and \(\omega\) (omega) is the angular frequency, which is related to how fast the oscillations occur. The general solution of this type of equation can be expressed using trigonometric functions like sine and cosine, which naturally represent cyclic phenomena due to their periodic nature.

• The harmonic oscillator's equation is key to understanding a wide variety of physical systems.
• Both mechanical and electrical systems can be modeled using this framework.
• These equations show how the position of a mass changes over time under a linear restoring force.

Harmonic oscillators are not only theoretical constructs but are found throughout science and engineering, illustrating rhythm and regularity in the natural world.

Initial Value Problems

Initial value problems are a fundamental concept in the study of differential equations. They are problems where we seek a specific solution to a differential equation that satisfies certain given conditions at a particular point in time. For our example, we have initial conditions \(x(0) = x_0\) and \(x'(0) = x_1\) applied to the differential equation \(x'' + \omega^2 x = 0\).

• The goal of solving an initial value problem is to find a unique function that satisfies both the equation and the initial conditions.
• Initial conditions like these help us pinpoint the exact solution or path needed from a family of possible solutions.
• Once initial conditions are applied, you can pinpoint how a system starts and evolves over time.

In practical terms, initial value problems allow us to determine how systems behave not just in general, but specifically, giving us powerful predictive capabilities.

Trigonometric Solutions

Trigonometric functions such as sine and cosine are intrinsic to the solution of second-order differential equations, especially those involving oscillatory behavior like the harmonic oscillator. The function \[ x(t) = c_1 \cos \omega t + c_2 \sin \omega t \] represents the general solution of our harmonic oscillator equation. These functions are periodic, making them perfect for modeling repeated behaviors over time such as vibrations, waves, and oscillations.

• Cosine and sine functions effectively capture the essence of oscillations due to their wave-like nature.
• Constants \(c_1\) and \(c_2\) are determined by the initial conditions, allowing for a tailored fit to given scenarios.
• Trigonometric solutions demonstrate how oscillations can be timed precisely using their frequencies and amplitudes.

In many engineering tasks, finding these exact solutions is vital to predicting how an oscillating system can operate under real-world constraints and adjustments.