91Ó°ÊÓ

A system of differential equations is a collection of two or more interrelated differential equations. These are equations involving derivatives of unknown functions with respect to one or more variables. In our context, we are dealing with a system consisting of two second-order differential equations:

• \( \frac{d^2 x}{dt^2} = 4y + e^t \)
• \( \frac{d^2 y}{dt^2} = 4x - e^t \)

Here, the equations are linked because the second derivative of one variable involves the presence of the other variable. This interconnection illustrates how changes in one part of the system affect the other. Understanding these systems is crucial for modeling complex phenomena where multiple variables interact. This forms the backbone of systems like mechanical structures in engineering or predator-prey models in biology.

The second derivative of a function provides information about the curvature or concavity of the graph of the function. It is the derivative of the derivative, or in simpler terms, it tracks how the rate of change itself is changing. In our exercise, we are required to find the second derivatives of the functions \( x(t) \) and \( y(t) \):

• For \( x(t) = \cos 2t + \sin 2t + \frac{1}{5}e^t \), the second derivative is \( \frac{d^2 x}{dt^2} = -4\cos 2t - 4\sin 2t + \frac{1}{5}e^t \).
• Similarly, for \( y(t) = -\cos 2t - \sin 2t - \frac{1}{5}e^t \), the second derivative is \( \frac{d^2 y}{dt^2} = 4\cos 2t + 4\sin 2t - \frac{1}{5}e^t \).

These second derivatives are then used to verify if \( x(t) \) and \( y(t) \) satisfy the given differential equations.

Verification of solutions in systems of differential equations involves proving that given functions indeed satisfy the equations within the system. This process typically includes computing derivatives of the given functions and substituting them back into the differential equations to show that they hold true. For our problem:

• Substitute \( y(t) \) into \( 4y + e^t \) to check the first equation; we find that \( 4(-\cos 2t - \sin 2t - \frac{1}{5}e^t) + e^t = -4\cos 2t - 4\sin 2t + \frac{1}{5}e^t \), which aligns with \( \frac{d^2 x}{dt^2} \).
• Similarly, for the second equation, substituting \( x(t) \) into \( 4x - e^t \) results in \( 4(\cos 2t + \sin 2t + \frac{1}{5}e^t) - e^t = 4\cos 2t + 4\sin 2t - \frac{1}{5}e^t \), matching \( \frac{d^2 y}{dt^2} \).

This confirmation process is vital for ensuring correctness in mathematical modeling and other applications.

Mathematical modeling is the process of using mathematical concepts and structures to represent real-world systems. Such models help in predicting and understanding the behavior of complex systems. Differential equations, particularly systems of differential equations, are a powerful tool in mathematical modeling. These allow for the description of relationships across variables that evolve over time. In our exercise, the given differential equations could represent a system where two quantities (modeled by \( x(t) \) and \( y(t) \)) influence each other, like oscillations in coupled pendulums or electrical circuits:

• The variable \( x(t) \) might represent displacement, voltage, or some other quantity of interest.
• The variable \( y(t) \) could represent a related property, influenced by changes in \( x(t) \).

Through mathematical modeling, such systems can be better understood and manipulated, providing insights into phenomena across fields like physics, biology, and engineering.