Question

(a) State whether or not the equation is autonomous. (b) Identify all equilibrium solutions (if any). (c) Sketch the direction field for the differential equation in the rectangular portion of the \(t y\)-plane defined by \(-2 \leq t \leq 2,-2 \leq y \leq 2\). $$ y^{\prime}=y^{2}-y $$

Short answer

Identify all equilibrium solutions and sketch the direction field in the rectangular portion \(-2 \leq t \leq 2, -2 \leq y \leq 2\) of the \(ty\)-plane. Answer: The given differential equation is autonomous since it does not depend on the independent variable, \(t\). The equilibrium solutions are \(y = 0\) and \(y = 1\). The direction field can be sketched by dividing the domain into three regions and drawing arrows according to the behavior of \(y'\) in each region (upwards if \(y' > 0\) and downwards if \(y' < 0\)).

Step-by-step solution

Part (a): Determine if the equation is autonomous

An equation is autonomous if it does not depend explicitly on the independent variable, in our case \(t\). The given differential equation is \(y' = y^2 - y\). This equation does not involve \(t\) explicitly, only involving the dependent variable \(y\). Therefore, the equation is autonomous.

Part (b): Identify all equilibrium solutions

To find the equilibrium solutions, we need to set the derivative \(y'\) equal to zero and solve for \(y\). This will give us the solutions for which the rate of change of \(y\) with respect to \(t\) is constant. \(0 = y^2 - y\) This can be factored as follows: \(0 = y(y - 1)\) So, the equilibrium solutions are \(y = 0\) and \(y = 1\).

Part (c): Sketching the direction field

To sketch the direction field, we must examine the behavior of the function in the given domain and recall that the direction of the arrows depends on the value of \(y'\). For this rectangular portion \(-2 \leq t \leq 2, -2 \leq y \leq 2\), consider the following possibilities for \(y\): 1. \(-2 \leq y < 0:\) In this region, \(y^2 - y > 0\), which implies that \(y'(t) > 0\). So the arrows will point upwards. 2. \(0 < y < 1:\) In this region, \(y^2 - y < 0\), which implies that \(y'(t) < 0\). So the arrows will point downwards. 3. \(1 < y \leq 2:\) In this region, \(y^2 - y > 0\), which implies that \(y'(t) > 0\). So the arrows will point upwards. Now that we have the behavior for each region, we can sketch the direction field. Draw the three regions described above on the \(ty\)-plane with horizontal arrows at the equilibrium points \(y = 0\) and \(y = 1\) (since \(y' = 0\) at these points). In each region, draw arrows with the appropriate direction, upwards if \(y' > 0\) and downwards if \(y' < 0\). This will provide a visual representation of the behavior of the differential equation in the given domain.

Key concepts

Equilibrium Solutions

In the context of differential equations, equilibrium solutions represent situations where the rate of change of the system is zero. This means that the dependent variable, often denoted as \( y \), doesn't change over time. For our differential equation \( y' = y^2 - y \), we find equilibrium solutions by setting \( y' = 0 \). This leads to the equation \( y^2 - y = 0 \).To solve for \( y \), we factor the equation:

• \( y(y - 1) = 0 \)

This gives us two solutions, \( y = 0 \) and \( y = 1 \), indicating that these are the equilibrium points where the system remains constant.These equilibrium solutions tell us the states \( y \) can remain in without changing, despite changes in the independent variable (like \( t \)). They are crucial for understanding the long-term behavior of the differential system, as any initial condition will trend towards or away from these points depending on their stability.

Direction Field

A direction field is a visual tool used to understand and analyze differential equations. It provides a pictorial representation of the solutions by illustrating the slope of the solution curves at various points in the \( ty \)-plane.For the equation \( y' = y^2 - y \), consider the different regions:

• For \(-2 \leq y < 0\), \( y^2 - y > 0 \), indicating that the slope \( y' \) is positive, and arrows point upwards.
• For \(0 < y < 1\), \( y^2 - y < 0 \), indicating that \( y' \) is negative, and arrows point downwards.
• For \(1 < y \leq 2\), \( y^2 - y > 0 \), indicating \( y' \) is positive, and arrows point upwards.

In your sketch, place horizontal arrows at the equilibrium solutions \( y = 0 \) and \( y = 1 \), as \( y' = 0 \) there. By observing the direction of the arrows in various regions, we can gauge how solutions behave – whether they rise, descend, or remain steady. This makes the direction field extremely helpful for predicting the qualitative behavior of solutions without actually solving the differential equation.

Differential Equation Analysis

Analyzing a differential equation like \( y' = y^2 - y \) involves understanding the dynamics it describes. An **autonomous differential equation** is one where the rate of change of the system doesn't depend explicitly on the independent variable \( t \). This gives the system a consistent behavior over time irrespective of the moment \( t \) itself.By identifying equilibrium solutions and sketching the direction field, we delve into a deeper analysis of the system:

• **Behavior around Equilibria:** Solutions near \( y = 0 \) and \( y = 1 \) are examined based on \( y' \). Solutions that start near \( y = 0 \) will slightly move away or towards it depending on stability, whereas a solution near \( y = 1 \) might exhibit similar attractions or repulsions.
• **Global Behavior:** The signs of \( y' \) across regions not only indicate motion direction but can imply stability of equilibrium solutions and cyclic behaviors over the allowed \( y \) domain.

Differential equation analysis equips us with an understanding of what happens to \( y \) over time, helping to predict future states and recognize patterns in physical, biological, or economic contexts encoded by such equations.