Problem 12 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Show that the characteristic equation of $$ \begin{aligned} x_{n+1} &=p \cdot x_{n}-q \cdot y_{n} \\ y_{n+1} &=x_{n} \end{aligned} $$ is Exercise 16.2 .12 Show that the characteristic equation of $$ \begin{array}{l} x_{n+1}=p \cdot x_{n}-q \cdot y_{n} \\ y_{n+1}=x_{n} \end{array} $$ is $$ x_{n+2}-p x_{n+1}+q x_{n}=0 $$

Step-by-Step Solution

Answer

The characteristic equation is \( x_{n+2} - p x_{n+1} + q x_{n} = 0 \).

Step 1: Write the given system of equations

The given system of equations is: \( x_{n+1} = p \cdot x_{n} - q \cdot y_{n} \) and \( y_{n+1} = x_{n} \). This represents a linear system in terms of \( x_n \) and \( y_n \).

Step 2: Substitute for \( y_{n+1} \)

Using the second equation, substitute \( y_{n} = x_{n-1} \) into the first equation. We have \( x_{n+1} = p \cdot x_{n} - q \cdot x_{n-1} \).

Step 3: Find an expression for \( x_{n+2} \)

Apply the transformation \( y_{n+1} = x_{n} \) to find \( x_{n+2} \): from the equation \( x_{n+1} = p \cdot x_{n} - q \cdot x_{n-1} \), set \( x_{n+2} = p \cdot x_{n+1} - q \cdot x_{n} \).

Step 4: Substitute \( x_{n+1} \) into \( x_{n+2} \)

We know \( x_{n+1} = p \cdot x_{n} - q \cdot x_{n-1} \). Substitute this into the equation for \( x_{n+2} \):\[ x_{n+2} = p(p \cdot x_{n} - q \cdot x_{n-1}) - q \cdot x_{n} \].

Step 5: Simplify the expression for \( x_{n+2} \)

Simplify the expression derived in Step 4 to obtain the homogeneous linear recurrence relation:\[ x_{n+2} = p^2 \cdot x_{n} - pq \cdot x_{n-1} - q \cdot x_{n} \].

Step 6: Factor and rearrange terms

Combine terms to arrange the equation in the standard form:\[ x_{n+2} - p \cdot x_{n+1} + q \cdot x_{n} = 0 \].This is the characteristic equation of the given system.

Key Concepts

Linear Recurrence Relation

A linear recurrence relation is a sequence of numbers where each term is a linear combination of previous terms. In simpler terms, the next term in the sequence depends linearly on earlier terms. For this exercise, we're looking at a second-order recurrence relation, which means that each term is determined by the two preceding terms. For instance, in the given characteristic equation \( x_{n+2} - p \cdot x_{n+1} + q \cdot x_{n} = 0 \), each term is expressed in terms of the two preceding terms, \( x_{n+1} \) and \( x_{n} \). The constants \( p \) and \( q \) act as coefficients, influencing how much each of these terms contribute to the next term in the sequence. Linear recurrence relations are pivotal in mathematical modeling because they provide a straightforward method to study systems' behaviors, predict future terms, and solve complex problems by breaking them down into simpler, recursive steps.

Homogeneous Equations

Homogeneous equations play a central role in understanding linear recurrence relations. A homogeneous equation, quite simply, is one where the function equals zero for every value of the variables. In the context of our exercise, it provides a neat framework where all terms on one side of the equation can completely define the behavior of the system without additional input.For the equation \( x_{n+2} - p \cdot x_{n+1} + q \cdot x_{n} = 0 \),it is classified as a homogeneous equation because all terms involve the sequence \( x \) itself, with no extraneous components. This structure ensures that solutions involve basic repeated patterns, leading to predictable geometric growth or decay, which is a fundamental aspect of solving and understanding these types of linear recurrence relations.

System of Equations

A system of equations is a collection of two or more equations with a common set of unknowns. They are incredibly useful in solving complex problems where multiple conditions must be satisfied simultaneously. In our context, the system consists of two linear equations:

• \( x_{n+1} = p \cdot x_{n} - q \cdot y_{n} \)
• \( y_{n+1} = x_{n} \)

These equations describe how the variables \( x_n \) and \( y_n \) evolve with each step, affecting each other continuously. The interconnected relation requires solving them together to understand the sequence's progression and ultimately determine the characteristic equation. By transforming the system to achieve the form \( x_{n+2} - p \cdot x_{n+1} + q \cdot x_{n} = 0 \), we've reduced a system of equations into a single homogenous linear recurrence relation. This is a crucial step that transforms complex interdependent relationships into solvable mathematical formulas. Understanding how systems of equations work simplifies solving intricate layers of interdependencies in mathematical problems.