Problem 39 - Advanced Engineering Mathematics

Question

(a) Show that the solution of the initial-value problem $$ \begin{aligned} &\frac{d^{2} x}{d t^{2}}+\omega^{2} x=F_{0} \cos \gamma t, x(0)=0, x^{\prime}(0)=0 \\ &\text { is } \quad x(t)=\frac{F_{0}}{\omega^{2}-\gamma^{2}}(\cos \gamma t-\cos \omega t) \end{aligned} $$ (b) Evaluate \(\lim _{\gamma \rightarrow \infty} \frac{F_{0}}{\omega^{2}-\gamma^{2}}(\cos \gamma t-\cos \omega t)\).

Step-by-Step Solution

Answer

The solution is \( x(t) = \frac{F_0}{\omega^2 - \gamma^2}(\cos \gamma t - \cos \omega t) \) and the limit is 0.

Step 1: Identify the Differential Equation Type

Recognize that the given differential equation \( \frac{d^2 x}{dt^2} + \omega^2 x = F_0 \cos \gamma t \) is a second-order, non-homogeneous linear ordinary differential equation with constant coefficients.

Step 2: Solve the Homogeneous Equation

First, solve the homogeneous part of the differential equation \( \frac{d^2 x}{dt^2} + \omega^2 x = 0 \). This is a simple harmonic oscillator equation, whose general solution is \( x_h(t) = C_1 \cos(\omega t) + C_2 \sin(\omega t) \).

Step 3: Find a Particular Solution

To find a particular solution to the non-homogeneous equation, use the method of undetermined coefficients. Assume a solution of the form \( x_p(t) = A \cos(\gamma t) + B \sin(\gamma t) \), where \( A \) and \( B \) are constants to be determined by substituting into the differential equation and equating coefficients.

Step 4: Calculate the Particular Solution Coefficients

Compute \( A \) and \( B \) by plugging \( x_p(t) \) into the left-hand side of the differential equation and matching terms with the right-hand side \( F_0 \cos(\gamma t) \). This yields \( A = \frac{F_0}{\omega^2 - \gamma^2} \) and \( B = 0 \). Thus, \( x_p(t) = \frac{F_0}{\omega^2 - \gamma^2} \cos(\gamma t) \).

Step 5: Combine Homogeneous and Particular Solutions

The general solution of the differential equation is the sum of the homogeneous and particular solutions: \( x(t) = x_h(t) + x_p(t) = C_1 \cos(\omega t) + C_2 \sin(\omega t) + \frac{F_0}{\omega^2 - \gamma^2} \cos(\gamma t) \).

Step 6: Apply Initial Conditions

Use the initial conditions \( x(0) = 0 \) and \( x'(0) = 0 \) to determine the constants \( C_1 \) and \( C_2 \). This gives two equations: \( C_1 + \frac{F_0}{\omega^2 - \gamma^2} = 0 \) and \( C_2\omega = 0 \). Solving these, we get \( C_1 = -\frac{F_0}{\omega^2 - \gamma^2} \) and \( C_2 = 0 \).

Step 7: Finalize the Solution

Substitute \( C_1 \) and \( C_2 \) back into the general solution to obtain the specific solution: \[ x(t) = \frac{F_0}{\omega^2 - \gamma^2} (\cos(\gamma t) - \cos(\omega t)) \]. This verifies part (a) of the problem.

Step 8: Evaluate the Limit as \( \gamma \rightarrow \infty \)

Consider \( \lim_{\gamma \rightarrow \infty} \frac{F_0}{\omega^2 - \gamma^2} (\cos(\gamma t) - \cos(\omega t)) \). As \( \gamma \to \infty \), the term \( \omega^2 - \gamma^2 \) approaches a very large negative number, causing the fraction to converge to zero. Thus, the limit is 0.

Key Concepts

Non-homogeneous Differential Equations

Non-homogeneous differential equations are a vital part of calculus, often showcasing real-world applications. These equations differ from their homogeneous counterparts by having a non-zero function on the right side. In our exercise, the differential equation is \( \frac{d^{2} x}{d t^{2}} + \omega^{2} x = F_{0} \cos \gamma t \).
The term \(F_{0} \cos \gamma t\) makes it non-homogeneous, as this part is not dependent solely on the function being solved for, \(x(t)\).

To tackle such equations, a typical approach is to find the general solution, which is the sum of the homogeneous equation's solution and a particular solution to the non-homogeneous equation. This superposition principle allows us to separate the natural behavior of the system from any external force represented by the inhomogeneous term.
In simpler terms, you first ignore the \(F_0 \cos \gamma t\) and solve the resulting homogeneous equation. Then, address the non-homogeneous part to find a solution that satisfies the entire equation.

Initial Value Problem

Solving differential equations often comes with additional conditions known as "initial values." These are used to find the specific solution for a given initial state of the system. In the context of our problem, two initial conditions are given: \(x(0)=0\) and \(x'(0)=0\).
These conditions imply that initially, the function value and its first derivative (which represents velocity, if \(x(t)\) represents position) are both zero.

The initial values are critical because the general solution to a differential equation contains arbitrary constants. By applying the initial conditions, we derive a unique solution from the general one, effectively boiling down to pinpointing particular values for these constants. In mathematical terms, initial values make sure the solution "starts" from a specific point on its trajectory.

Second-order Linear ODEs

A second-order linear ordinary differential equation (ODE) is characterized by containing the second derivative of a function. Our equation \( \frac{d^2 x}{dt^2} + \omega^2 x = F_0 \cos \gamma t \) fits this definition perfectly. Here, the highest derivative present is the second derivative \(\frac{d^2 x}{dt^2}\).

This class of equations is particularly prevalent in physics and engineering, where they often model oscillatory systems like springs and circuits. The phrase 'linear' refers to the fact that the function \(x(t)\) and its derivatives only appear to the first power, and their coefficients are constant.

• Linear ODEs allow superposition, meaning solutions can be added together to form more general solutions.
• This property simplifies the solving process by breaking complex problems into manageable parts.

Working with these equations involves finding a complementary solution to the associated homogeneous equation and combining it with a particular solution tailored to the non-homogeneous part.

Method of Undetermined Coefficients

The method of undetermined coefficients is a systematic approach to find particular solutions of non-homogeneous linear differential equations. This method works particularly well when the non-homogeneous term (like \(F_0 \cos \gamma t\) in our problem) is a simple function such as a polynomial, exponential, sine, or cosine.
The method requires guessing a form for the particular solution with undetermined coefficients. In our example, a solution like \(x_p(t) = A \cos(\gamma t) + B \sin(\gamma t)\) is proposed.

Next, this form is substituted into the differential equation to determine the values of \(A\) and \(B\) that make it a solution. This involves matching coefficients with the non-homogeneous part, solving the resulting equations for \(A\) and \(B\).
This technique leverages the algebraic pattern of the differential equation and the nature of the non-homogeneous term to find straightforward solutions, avoiding more complex integral approaches.