Problem 38 - Advanced Engineering Mathematics

Question

In Problems 37 and 38 , verify that the indicated pair of functions is a solution of the given system of differential equations on the interval \((-\infty, \infty)\) $$ \begin{aligned} &\frac{d^{2} x}{d t^{2}}=4 y+e^{t} \\ &\frac{d^{2} y}{d t^{2}}=4 x-e^{t} \\ &x=\cos 2 t+\sin 2 t+\frac{1}{5} e^{t} \\ &y=-\cos 2 t-\sin 2 t-\frac{1}{5} e^{t} \end{aligned} $$

Step-by-Step Solution

Answer

The functions are solutions to the differential equations on \((- fty, fty)\) as shown.

Step 1: Differentiate x and y Twice

The given functions are \( x = \cos 2t + \sin 2t + \frac{1}{5} e^{t} \) and \( y = -\cos 2t - \sin 2t - \frac{1}{5} e^{t} \). We need to find the second derivatives \( \frac{d^2 x}{d t^2} \) and \( \frac{d^2 y}{d t^2} \).First, find the first derivatives:- \( \frac{d x}{d t} = -2\sin 2t + 2\cos 2t + \frac{1}{5} e^{t} \)- \( \frac{d y}{d t} = 2\sin 2t - 2\cos 2t - \frac{1}{5} e^{t} \)Then, find the second derivatives:- \( \frac{d^2 x}{d t^2} = -4\cos 2t - 4\sin 2t + \frac{1}{5} e^{t} \)- \( \frac{d^2 y}{d t^2} = 4\cos 2t + 4\sin 2t - \frac{1}{5} e^{t} \).

Step 2: Plug into Differential Equations

Now substitute \( \frac{d^2 x}{d t^2} \) and \( \frac{d^2 y}{d t^2} \) into the original differential equations to check for consistency.For the equation \( \frac{d^2 x}{d t^2} = 4y + e^{t} \):- Left Side: \( \frac{d^2 x}{d t^2} = -4\cos 2t - 4\sin 2t + \frac{1}{5} e^{t} \).- Right Side: \( 4(-\cos 2t - \sin 2t - \frac{1}{5} e^{t}) + e^{t} = -4\cos 2t - 4\sin 2t - \frac{4}{5} e^{t} + e^{t} = -4\cos 2t - 4\sin 2t + \frac{1}{5} e^{t} \).Both sides are equal, thus verifying the first equation.

Step 3: Verify Second Differential Equation

For the equation \( \frac{d^2 y}{d t^2} = 4x - e^{t} \):- Left Side: \( \frac{d^2 y}{d t^2} = 4\cos 2t + 4\sin 2t - \frac{1}{5} e^{t} \).- Right Side: \( 4(\cos 2t + \sin 2t + \frac{1}{5} e^{t}) - e^{t} = 4\cos 2t + 4\sin 2t + \frac{4}{5} e^{t} - e^{t} = 4\cos 2t + 4\sin 2t - \frac{1}{5} e^{t} \).Both sides are equal, confirming the second equation.

Key Concepts

System of Differential Equations

A system of differential equations is a collection of two or more related differential equations involving more than one dependent variable. In this context,

• The system connects variables through derivatives.
• It often arises in modeling scenarios where multiple quantities are interrelated.

For instance, in the exercise, there's a pair of equations:1. \( \frac{d^2 x}{dt^2} = 4y + e^t \)2. \( \frac{d^2 y}{dt^2} = 4x - e^t \)
The goal is to find functions \( x(t) \) and \( y(t) \) that satisfy both equations simultaneously over a specified interval.
Systems like these frequently model oscillatory or coupled behaviors in physics and engineering, such as the motion of coupled pendulums or electrical circuits.

Second Derivative

The second derivative of a function provides information about its curvature, or how it changes shape. It can indicate points of concavity or convexity.

• If the second derivative is positive, the function is concave up.
• If negative, it is concave down.

Specifically, for functions \( x(t) \) and \( y(t) \) given in the exercise:

• We compute \( \frac{d^2 x}{dt^2} \) and \( \frac{d^2 y}{dt^2} \) to check the behavior of the solutions.
• A crucial part of solving the system involves substituting these second derivatives into the original differential equations.

In this case, differentiating each function twice provides the expressions necessary to verify the system.

Verification of Solutions

Verification is a critical process that confirms whether a proposed solution is indeed correct for a given differential equation. It involves substituting the function and its derivatives into the original equations to verify consistency. Such a step ensures

• The proposed functions \( x(t) \) and \( y(t) \) truly satisfy the differential equations across the intended interval.
• Errors in derivation or application of the system are identified.

For this exercise, after calculating the second derivatives, the next steps involved matching both sides of the differential equations. This match thus confirms that the proposed functions are accurate solutions for the system.

Trigonometric Functions

Trigonometric functions like sine and cosine are periodic and often appear in differential equations. They model oscillating behaviors such as sound waves and pendulum motions. In the exercise,

• Both \( x(t) \) and \( y(t) \) involve combinations of \( \cos(2t) \) and \( \sin(2t) \) reflecting periodic aspects of the solution.
• They also add exponential terms to handle non-periodic changes, as evident in the presence of the term \( \frac{1}{5}e^t \).

These functions are essential because they represent repeating cycles, critical in various physical phenomena, making them common elements in differential equations.