Question

What can you say about the behavior of the solution? Are there equilibria? What stability do these equilibria have? $$ y^{\prime}=y^{2}-1 $$

Short answer

There are two equilibria at \( y = 1 \) (unstable) and \( y = -1 \) (stable). Solutions near \( y = 1 \) diverge, while they converge to \( y = -1 \).

Step-by-step solution

Identify the Equilibria

To find the equilibria, set the derivative function to zero, i.e., \( y' = 0 \). Therefore, solve \( y^2 - 1 = 0 \) to find \( y = 1 \) and \( y = -1 \). These are the equilibrium points of the differential equation.

Analyze Stability using Linearization

To assess stability, we need the derivative of the equation \( dy/dt = y^2 - 1 \) with respect to \( y \). The derivative is \( f'(y) = 2y \). Evaluate it at each equilibrium point. At \( y = 1 \), \( f'(1) = 2 \times 1 = 2 \), and at \( y = -1 \), \( f'(-1) = 2 \times (-1) = -2 \).

Interpret Stability of Each Equilibrium

An equilibrium point is stable if \( f'(y) < 0 \) and unstable if \( f'(y) > 0 \). At \( y = 1 \), since \( f'(1) = 2 > 0 \), the equilibrium is unstable. At \( y = -1 \), since \( f'(-1) = -2 < 0 \), the equilibrium is stable.

Describe Overall Behavior of Solutions

The solutions evolve based on initial values: they move away from the unstable point \( y = 1 \) and towards the stable point \( y = -1 \). If \( y(0) > 1 \), \( y \) grows without bound; if \(-1 < y(0) < 1\), \( y \) decreases to \(-1\).

Key concepts

Equilibrium Points

Equilibrium points in differential equations are values where the derivative is zero, indicating that the change rate at that point is zero. This means that if the function reaches this point, it will remain constant. To find the equilibrium points, we set the equation's derivative to zero. In our case, we solve the equation \( y^2 - 1 = 0 \). This gives us the solutions \( y = 1 \) and \( y = -1 \). These solutions are our equilibrium points, meaning if the system reaches these values, it will stay there unless disturbed.

Stability Analysis

Stability analysis helps us determine whether a system will return to equilibrium after a small disturbance. We can determine stability by calculating the derivative of the differential equation concerning \( y \). Here, we have \( f(y) = y^2 - 1 \), and its derivative is \( f'(y) = 2y \).
If the derivative at an equilibrium point is less than zero (\( f'(y) < 0 \)), the equilibrium is stable; it signifies that the system will restore itself back to equilibrium if slightly disturbed.
Conversely, if the derivative is more than zero (\( f'(y) > 0 \)), the equilibrium is unstable and the system will diverge away from equilibrium when disturbed. At \( y = 1 \), \( f'(1) = 2 \), indicating instability as it is greater than zero. However, at \( y = -1 \), \( f'(-1) = -2 \), showing stability due to the negative value.

Linearization

Linearization is a method for approximating the behavior of a nonlinear system around its equilibrium points by using a linear equation. This involves using the first derivative or the Jacobian matrix for a single variable.For our differential equation \( y' = y^2 - 1 \), the linearization process required taking the derivative \( f'(y) = 2y \).
This derivative helps us understand how small perturbations around an equilibrium point will affect the system. Specifically, it tells us about the speed and direction of the system’s response to this perturbation, allowing us to make informed predictions about stability and behaviour.

Solution Behavior

The behavior of the solutions of a differential equation depends on initial conditions and the stability of its equilibrium points. For our differential equation, initial conditions determine whether solutions head towards or away from equilibrium points.

• If an initial point is greater than the unstable equilibrium \( y = 1 \), solutions move away and may grow indefinitely.
• For initial points between \( -1 \) and \( 1 \), solutions tend to move towards the stable equilibrium \( y = -1 \).
• This kind of solution behavior helps forecast the long-term behavior of a system.

Understanding these aspects of solution behavior is critical for predicting how a system will evolve over time based on its initial state.