Question

What is the equilibrium solution of the equation $y^{\prime }(t)=3y-9?$ Is it stable or unstable?

Short answer

Answer: The equilibrium solution is y = 3, and its stability is semi-stable.

Step-by-step solution

Find the equilibrium solution

Set y'(t) to zero and solve for y: $$0=3y-9$$ Divide both sides by 3: $$y=3$$ The equilibrium solution is y = 3.

Analyze stability

To analyze stability, we look at how the solution behaves around the equilibrium point. In this case, we will consider the function y'(t) - 3y + 9: 1. If the expression is positive for y > 3 and negative for y < 3, the equilibrium is stable. 2. If the expression is negative for y > 3 and positive for y < 3, the equilibrium is unstable. 3. If the expression is always zero, the equilibrium is semi-stable. We have: $$y^{\prime }(t)-3y+9=3y-9-3y+9=0$$ In this case, the expression is always zero, which means the equilibrium solution y = 3 is semi-stable.

Key concepts

Differential Equation

A differential equation is a mathematical relation that involves the derivatives of a function. It shows how a function changes with respect to another variable, usually time or space. In simpler terms, it's like a recipe that tells you how a situation evolves or how a system behaves under certain conditions. Differential equations can describe a wide range of phenomena, from bouncing balls to population dynamics. They're like unstoppable story tellers for nature’s scripts.

In regards to the exercise, the equation given is $y^{\prime }(t)=3y-9$, which is a first-order linear differential equation. First-order because it involves the first derivative $y^{\prime }(t)$, and linear because none of the terms are raised to a power or multiplied by each other.

Understanding their structure and behavior helps us to unravel the equilibrium solutions.

Stability Analysis

Stability analysis is an essential part of understanding how systems behave over time, especially near an equilibrium solution. Stability investigates whether small deviations from equilibrium will result in movements back to, away from, or neither, keeping the system understood and predictable.

To analyze stability, we look at the expression $3y-9$. Around the equilibrium point $y=3$, we evaluate the slope or direction of the system. If changes push the value towards the equilibrium in both directions, it's stable. If they push away, it’s unstable.

In the exercise, the expression $y^{\prime }(t)=3y-9$ always evaluates to zero when analyzed around $y=3$. Therefore, the equilibrium solution is neither stable nor unstable, finding itself as semi-stable. This means that infinitesimally near deviations neither grow larger nor shrink back strongly.

Mathematical Modeling

Mathematical modeling is the process of translating problems from 'words and context' into mathematical language called models. These models help to simplify real-world complexities into understandable and comparable representations.

These mathematical blueprints can apply to economics, biology, physics, and beyond. Effective modeling involves using differential equations to forecast changes under set conditions, such as the velocity of a moving car, or how a population may grow or decline with time.

In this context, the differential equation $y^{\prime }(t)=3y-9$ represents a simplification of a dynamic system where the variable $y$ changes over time, driven by conditions expressed in the equation. Solving this provides an equilibrium, helping to predict or simulate potential real-life situations accurately. Thus, mastering these concepts equips learners with the tools to anticipate and evaluate systemic behaviors.

• Recognizes complex systems.
• Builds bridges between theoretical and practical scenarios.
• Increases the understanding of dynamic interactions.