Problem 20 - A First Course in Differential Equations with Modeling Applications

Question

Solve the given system of differential equations by systematic elimination. $$\begin{aligned} &\frac{d x}{d t}=-x+z\\\ &\frac{d y}{d t}=-y+z\\\ &\frac{d z}{d t}=-x+y \end{aligned}$$

Step-by-Step Solution

Answer

The system can be solved with the relationship \( z = y - x + C \).

Step 1: Label the Differential Equations

Let's label the given system of differential equations for clarity:1. \( \frac{dx}{dt} = -x + z \) 2. \( \frac{dy}{dt} = -y + z \) 3. \( \frac{dz}{dt} = -x + y \). These are the original equations we will work with.

Step 2: Express x and y in terms of z

From equations (1) and (2), we can express \(x\) and \(y\) in terms of \(z\). - From (1): \( x = z - \frac{dx}{dt} \)- From (2): \( y = z - \frac{dy}{dt} \).

Step 3: Substitute x and y into the third equation

Substitute the expressions for \(x\) and \(y\) into equation (3):\( \frac{dz}{dt} = -(z - \frac{dx}{dt}) + (z - \frac{dy}{dt}) \).This simplifies to \( \frac{dz}{dt} = \frac{dy}{dt} - \frac{dx}{dt} \).

Step 4: Integrate to Find Relationship among Derivatives

We integrate the equation \( \frac{dz}{dt} = \frac{dy}{dt} - \frac{dx}{dt} \) with respect to \(t\) to find:\( z = y - x + C \), where \(C\) is a constant of integration.

Step 5: Use the Relationship and Solve for z

Using \( z = y - x + C \), substitute back into the expressions for \(x\) and \(y\) obtained earlier:- \( x = y - x + C - \frac{dx}{dt} \),- \( y = y - x + C - \frac{dy}{dt} \).These expressions allow us to solve for the constant \(C\) and determine the values of \(x\) and \(y\) once initial conditions are provided.

Key Concepts

Systematic Elimination

Systematic elimination is a powerful technique often used to simplify and solve systems of equations. This method allows you to isolate variables step-by-step, making it easier to solve for unknowns. In the case of differential equations, systematic elimination helps streamline the solution process by reducing the number of equations in our system. Here, we strategically remove variables by expressing them in terms of others, simplifying the system along the way. This process involves making algebraic manipulations and substitutions repeatedly until we obtain a solution in terms of the remaining variables.

System of Equations

A system of equations consists of multiple equations that share common variables. In this exercise, we're dealing with a system of differential equations that involve the variables \(x\), \(y\), and \(z\). Each equation gives information about how one variable changes with respect to another. By combining these equations, we can gather comprehensive insights about the interplay between \(x\), \(y\), and \(z\). Thus, solving a system of equations often involves finding a suitable expression or value for each variable that simultaneously satisfies all equations in the system.

Integration

Integration is the inverse process of differentiation. It is a fundamental concept in calculus used to determine the original function from its derivative. In this exercise, after simplifying the system of equations, we reach an expression for the derivative of \(z\). To find \(z\) itself, we integrate this relationship with respect to time \(t\). The integration step helps us construct a relationship between the variables, bringing us closer to a solution. Don't forget, integration can introduce a constant term, represented here as \(C\), which must be determined based on additional information or initial conditions.

Initial Conditions

Initial conditions are specific values provided for variables at the beginning of the observation period. These conditions are vital for solving differential equations as they help identify the particular solution of a system out of the many possible general solutions. In our exercise, once we have the expression for \(z\) and its relationship to \(x\) and \(y\), the initial conditions allow us to determine the constant of integration \(C\). By applying these known values, we ensure that the system of equations accurately describes the behavior of the variables in their real-world context.