91Ó°ÊÓ

Differentiation is a powerful mathematical tool used to find the rate at which a function changes. When we differentiate a function of one variable with respect to another, we are essentially measuring how much the function's value is increasing or decreasing as the variable changes.
In the given problem, we start with the functions \(x = -e^t\) and \(y = e^t\). We differentiate each of these functions with respect to \(t\):- For \(x = -e^t\), the derivative \( \frac{dx}{dt} \) is \(-e^t\), which tells us how \(x\) changes as \(t\) changes.- For \(y = e^t\), the derivative \( \frac{dy}{dt} \) is \(e^t\), which similarly tells us how \(y\) changes with \(t\).This process of differentiation helps us to understand the behavior of the system by knowing the instantaneous rates of change of \(x\) and \(y\) with respect to time \(t\).

A system of equations is a set of equations with multiple variables that we aim to solve simultaneously. These systems are quite common in mathematical models that describe real-world phenomena. Understanding how variables are interrelated through equations helps predict the system's behavior.In the problem context, we are looking at a first-order system of two equations:- \( \frac{dx}{dt} = -y \)- \( \frac{dy}{dt} = -x \)These equations define how \(x\) and \(y\) interact and change with respect to time \(t\). Solving this system involves confirming solutions that satisfy both equations. This process is an essential part of mathematical modeling, allowing us to describe dynamic systems where each equation influences others within the system.

Solutions verification involves checking whether our proposed solutions correctly satisfy the given equations of the system. It's like a puzzle, where you have to make sure all the pieces fit together to form a coherent picture.To verify the solutions for our system:- We start by substituting \( \frac{dx}{dt} = -e^t \) and \(y = e^t\) into the equation \( \frac{dx}{dt} = -y \). Doing this gives us \(-e^t = -e^t\), which confirms that the solution fits.- Next, we substitute \( \frac{dy}{dt} = e^t \) and \(x = -e^t\) into the equation \( \frac{dy}{dt} = -x \). This gives us \(e^t = e^t\), which also holds true, completing our verification.Verification is crucial because it ensures that the solutions not only solve each equation independently, but also collectively fit the overall system of equations, thus maintaining the integrity of the mathematical model.