Question

Find the equilibrium solution of the following equations, make a sketch of the direction field, for \(t \geq 0,\) and determine whether the equilibrium solution is stable. The direction field needs to indicate only whether solutions are increasing or decreasing on either side of the equilibrium solution. $$y^{\prime}(t)-\frac{y}{4}-1=0$$

Short answer

#Question# Determine the equilibrium solution and its stability for the equation \(y'(t) = \frac{y}{4} + 1\) for \(t \geq 0\). #Answer# The equilibrium solution is \(y = 4\) and it is stable.

Step-by-step solution

Set \(y'(t)\) to zero.

To find the equilibrium solution, we need to set the derivative of y with respect to t to zero and solve for y. $$0 = y'(t) - \frac{y}{4} - 1$$ ##Step 2: Solving for y##

Rearrange and solve for y.

We will now rearrange and solve the equation for y to find the equilibrium solution. $$\frac{y}{4} = 1$$ $$y = 4$$ The equilibrium solution is \(y=4\). ##Step 3: Analyzing the direction field##

Analyze how \(y'(t)\) changes.

We need to analyze the behavior of solutions around the equilibrium solution (\(y=4\)). Check whether solutions are increasing or decreasing on either side of the equilibrium solution by finding the sign of \(y'(t)\) when \(y>4\) and \(y<4\). 1. When \(y>4\), \(y'(t) - \frac{y}{4} - 1 > 0\), which means \(y'(t)>0\), and solutions are increasing. 2. When \(y<4\), \(y'(t) - \frac{y}{4} - 1 < 0\), which means \(y'(t)<0\), and solutions are decreasing. ##Step 4: Sketch the direction field##

Sketch the direction field around the equilibrium solution.

Based on the analysis of the direction field in the previous step, we sketch the direction field around the equilibrium solution \(y=4\). Solutions are increasing when \(y>4\) and decreasing when \(y<4\). ##Step 5: Determine the stability of the equilibrium solution##

Determine the stability.

Based on the direction field pattern, since the solutions close to the equilibrium solution seem to converge towards the equilibrium solution, we can conclude that the equilibrium solution \(y=4\) is stable.

Key concepts

Direction Field

A direction field, also known as a slope field, is a visual representation used to understand the behavior of solutions to a first-order differential equation. It consists of segments or arrows at various points in the plane, each showing the slope of the solution passing through that point. By examining a direction field, one can qualitatively predict the behavior of solutions without solving the equation analytically.

For example, in our exercise, the directions at each point show whether the solution curve, at that point, is increasing or decreasing. This is determined by the sign of the derivative of the solution. When we sketch the direction field for the equation provided, one can notice that the arrows point upwards where the solution is increasing (\(y'>0\)) and downwards where the solution is decreasing (\(y'<0\)). Aligning these arrows with the given differential equation creates a pattern that visually assists in understanding the flow and behavior of the solution across different values of y.

Differential Equation

A differential equation is a mathematical equation that relates some function with its derivatives. In its most basic form, it represents a relationship between an unknown function and its rate of change. In the context of our exercise, the differential equation given is \(y'(t)-\frac{y}{4}-1=0\).

Such equations are fundamental in describing various phenomena in the sciences and engineering, including motion, growth, and decay, among others. Solving a differential equation provides a function that predicts the behavior of the system described by the equation. In our exercise, setting up and solving the equation reveals the equilibrium solution—the point where the rate of change of the function is zero, implying no further growth or decay.

Equilibrium Solution

An equilibrium solution to a differential equation is a solution that does not change with time. It corresponds to the condition where the derivative (or slope) of the function is zero, indicating a steady-state or balance. In the step-by-step solution, we find the equilibrium solution by solving \(\frac{y}{4} = 1\), giving us \(y = 4\), which is a constant solution.

Understanding equilibrium solutions is crucial as they represent states that the system can theoretically maintain indefinitely. In various scenarios, these can be points of stability (like resting positions in mechanics) or points where different forces are in balance (like the population number in a simple biological model).

Stability Analysis

Stability analysis in the context of differential equations involves determining whether small perturbations or changes to the equilibrium solution will cause the system to return to equilibrium or deviate further away. When a system returns to equilibrium after a small disturbance, the equilibrium is said to be 'stable'. Conversely, if it moves away, the equilibrium is referred to as 'unstable'.

In our exercise, the system's return to the equilibrium solution of \(y=4\) upon disturbance implies stability. This inference is deduced through the observation that solutions are decreasing when \(y>4\) (moving back towards \(y=4\)) and increasing when \(y<4\) (again moving towards \(y=4\)). This type of stability, known as 'asymptotic stability', is essential for understanding the long-term behavior of systems described by differential equations.