Problem 26 - Calculus

Question

which initial value produces \(y_{1}=y_{0}\) (steady state)? $$ y(t+1)=-\frac{1}{4} y(t)+6 $$

Step-by-Step Solution

Answer

The initial value is \(y_0 = 4.8\).

Step 1: Understand the Problem

The problem asks for the initial value that produces a steady state, meaning that \(y_{1} = y_{0}\). This indicates that the system does not change over time, so \(y(t + 1) = y(t)\).

Step 2: Set Up the Steady State Equation

We know that at steady state, \(y(t + 1) = y(t)\). Substitute this equality into the equation: \ \[ y(t) = -\frac{1}{4} y(t) + 6 \].

Step 3: Solve for the Steady State Value

Rearrange the equation from Step 2 to solve for \(y(t)\): \ \[ y(t) + \frac{1}{4}y(t) = 6 \]. Factor the left-hand side: \ \[ \frac{5}{4} y(t) = 6 \]. Divide each side by \(\frac{5}{4}\) to isolate \(y(t)\): \ \[ y(t) = \frac{6 \cdot 4}{5} = \frac{24}{5} \].

Step 4: Interpret the Solution

The solution \( y(t) = \frac{24}{5} \) or \(4.8\) is the initial value that produces a steady state. This means that if the initial value \(y_{0}\) is \(4.8\), then \(y_{1} = y_{0} = 4.8\).

Key Concepts

Steady State

In the context of difference equations, a steady state refers to a condition where the value of a sequence remains constant over time. Essentially, the sequence does not undergo any changes with each progression. For this to occur, the terms of the sequence must satisfy a particular condition where each subsequent value equals the previous one.
This can be expressed mathematically as:

• \( y(t+1) = y(t) \)

A steady state is significant because it represents a point of balance or equilibrium within the system. In practical terms, reaching a steady state may reflect stability or predictability in a model, such as a population level or economic indicator.
Identifying a steady state involves setting the equation for the sequence equal to its prior value and solving for the unknown variable, which in this problem is the initial value. Thus, for the given equation, the steady state arises when the initial value is 4.8, ensuring the sequence remains constant.

Initial Value

The initial value in a recursive sequence is the starting point from which the sequence begins to unfold. This initial condition is vital because it can determine the future behavior of the entire sequence.
In our exercise, the initial value is denoted as \( y_0 \) and the task is to find this specific value that leads directly into a steady state situation, where consecutive terms remain unchanged.

• For the equation \( y(t+1) = -\frac{1}{4} y(t) + 6 \), identifying the proper initial value ensures that the recursive formula holds true, maintaining equal subsequent terms.

In this instance, solving for the steady state value enabled us to determine that if \( y_0 \) is set to 4.8, the sequence will not change. This highlights how crucial it is to precisely define the starting point in such sequences.

Recursive Sequences

Recursive sequences are sequences where the terms are defined in relation to previous terms using a specific formula or rule. This recursive relation relies heavily on initial conditions, as once given, they enable the generation of subsequent terms.
Recursive sequences follow a format such as \( y(t+1) = f(y(t)) \), where each term is calculated by applying a function to the preceding term. This makes them distinct from other sequences where each term is calculated independently of the others.

• These sequences often represent dynamic systems, modeling real-world phenomena like population growth or financial forecasts.
• They can be simple or quite sophisticated, depending on the functional relation used.

In our problem, the recursive formula given is \( y(t+1) = -\frac{1}{4} y(t) + 6 \), demonstrating the idea that the value at the next time step depends on both the previous value and the specific rule governing change. Recognizing the structure of recursive sequences enables a deeper understanding of how systems evolve over time.