Problem 25 - Calculus

Question

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

Step-by-Step Solution

Answer

The initial value that produces a steady state is 3.

Step 1: Understand the Equation

We are given a difference equation: \( y(t+1) = -y(t) + 6 \). We need to find an initial value for \( y(0) \) that results in a steady state where \( y_{1} = y_{0} \). This means the output of the equation does not change with time.

Step 2: Set up the Steady State Equation

For the steady state, you want \( y(t+1) = y(t) \). Substitute \( y(t) \) with \( y(0) \) and \( y(t+1) \) with \( y(0) \) as well, so we have: \( y(0) = -y(0) + 6 \).

Step 3: Solve for the Initial Value

Solve the equation from the previous step: \( y(0) = -y(0) + 6 \). Add \( y(0) \) to both sides to get: \( 2y(0) = 6 \). Divide both sides by 2 to isolate \( y(0) \): \( y(0) = 3 \).

Key Concepts

Initial Value

In difference equations, the initial value is the starting point of a sequence or a function. It represents the first term or the beginning state of the system you're considering. Understanding the initial value is crucial when dealing with these types of problems because it sets the stage for how the system will evolve over time.

• The initial value is often denoted as \( y(0) \).
• It's the point at which you "kick off" the calculations for the rest of the sequence.
• Finding the correct initial value can ensure that the system behaves in a desired way, such as reaching a steady state.

In the original exercise, we were asked to find an initial value that would produce a steady state. This required setting up the equation in such a way that the future terms would match the initial one. By solving \( 2y(0) = 6 \), we determined that the ideal initial value was 3. This initial setup ensures that at each step \( y(t) = y(t+1) \), maintaining the steady state.

Steady State

A steady state in a difference equation is a condition where the values stop changing, meaning that once the system reaches this state, it continues to repeat the same value indefinitely. Understanding how and why systems reach a steady state is crucial in many fields, including engineering and economics.

• In mathematical terms, a steady state means \( y(t+1) = y(t) \).
• This implies that the system has reached equilibrium and will no longer oscillate or change—it's stable.
• Finding a steady state can help predict long-term behaviors of a system.

In this particular exercise, we set up the equation \( y(t+1) = -y(t) + 6 \) to match the condition of the steady state, resulting in the equation \( y(0) = -y(0) + 6 \). By solving this simple algebraic expression, we found that the steady state occurs when the initial value is 3.

Difference Equations in Calculus

Difference equations are vital in calculus as they help represent discrete changes rather than continuous. They play a significant role in modeling scenarios where changes occur at specific intervals—useful in digital processes and systems.

• They are the discrete counterparts of differential equations but deal with sequences rather than continuous functions.
• Difference equations analyze how a quantity changes from one period to another.
• They help in understanding the progression and evolution of systems over time.

In this exercise, we used a simple linear difference equation \( y(t+1) = -y(t) + 6 \). Through each step, we uncovered how the system needed adjustment, particularly finding the right initial value to reach a steady state. This kind of analysis falls under the umbrella of discrete dynamical systems and provides insights into behaviors modeled over specific, discrete periods.