Problem 24 - Calculus

Question

which initial value produces \(y_{1}=y_{0}\) (steady state)? $$ y(t+1)=\frac{1}{2} y(t)-6 $$

Step-by-Step Solution

Answer

The initial value that produces a steady state is \( y_0 = -12 \).

Step 1: Understand the Steady State Condition

The steady state condition is when the future value of the function is equal to its current value. This means that for the steady state, we have \( y_{t+1} = y_t \). We need to find the value of \( y_0 \) such that \( y_1 = y_0 \).

Step 2: Set up the Equation for Steady State

To find the steady state value, substitute \( y(t+1) = y(t) \) into the given function formula:\[ y_t = \frac{1}{2}y_t - 6 \]

Step 3: Solve the Equation for y(t)

Simplify the equation: \[ y_t = \frac{1}{2}y_t - 6 \]Subtract \( \frac{1}{2}y_t \) from both sides to isolate the variable:\[ y_t - \frac{1}{2}y_t = -6 \]This simplifies to:\[ \frac{1}{2}y_t = -6 \]

Step 4: Solve for y(t)

Multiply both sides of the equation by 2 to solve for \( y_t \):\[ y_t = -12 \]

Step 5: Verify the Solution

Substitute \( y_0 = -12 \) back into the original equation to verify it satisfies the steady state condition:\[ y(t+1) = \frac{1}{2}(-12) - 6 = -6 - 6 = -12 \]This confirms that \( y_1 = y_0 = -12 \).

Key Concepts

Understanding Discrete Dynamical Systems

Discrete dynamical systems are mathematical models used to describe the behavior of a system that evolves in discrete time steps. These systems are particularly useful for studying scenarios where changes happen at distinct intervals, for instance, yearly population changes or monthly financial forecasts. The core idea is that the state of the system at any given time depends on its state at a previous time.

• Time evolves in fixed steps, denoted as \( t, t+1, t+2, \) and so on.
• The state of the system at time \( t \) is represented by \( y(t) \).
• By employing a rule or equation, the future state \( y(t+1) \) can be calculated.

Discrete dynamical systems provide a simplified way to model complex systems using iterations. Each iteration applies a deterministic rule, allowing us to predict future states based on initial conditions and parameters of the system. This is precisely what we're observing with our steady state analysis, where we aim to find a state that remains constant over time.

Exploring Difference Equations

A difference equation is a mathematical expression involving the differences between successive terms of a sequence. These equations relate the values of a sequence at different times and are fundamental in analyzing discrete dynamical systems.

• Difference equations are similar to differential equations, but the for discrete rather than continuous changes.
• They often model phenomena in economics, biology, and other fields where discrete change is observed.

For example, in our exercise, the equation \( y(t+1) = \frac{1}{2} y(t) - 6 \) describes how the quantity \( y \) changes from one time step to the next. Solving these equations often involves finding explicit formulas for the sequences. Here, we solve for \( y \) such that the sequence reaches a steady state, meaning \( y(t+1) = y(t) \), which simplifies our calculations significantly.

Solving the Initial Value Problem

An initial value problem (IVP) is a type of problem that seeks to find a sequence of values given an initial condition. In order to determine the future behavior of a sequence in a discrete dynamical system, we typically require a starting point or initial value.

• The initial value, \( y_0 \), is the known starting condition of the system.
• With this value, and using the difference equation, we can predict the future states of the system.

In our scenario, we are looking to find the initial value, \( y_0 \), that results in a steady state. This means that no matter the time step, the value of \( y \) remains constant. By setting \( y_1 = y_0 \) in our equation, we derive that \( y_0 = -12 \), satisfying the condition of maintaining a steady state throughout the dynamics of the system. This illustrates the essence of solving initial value problems within discrete models.