Question

Equilibrium solutions \(A\) differential equation of the form \(y^{\prime}(t)=f(y)\) is said to be autonomous (the function \(f\) depends only on y. The constant function \(y=y_{0}\) is an equilibrium solution of the equation provided \(f\left(y_{0}\right)=0\) (because then \(y^{\prime}(t)=0\) and the solution remains constant for all \(t\) ). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations. a. Find the equilibrium solutions. b. Sketch the direction field, for \(t \geq 0\). c. Sketch the solution curve that corresponds to the initial condition \(y(0)=1\). $$y^{\prime}(t)=y(2-y)$$

Short answer

Based on the given information and step by step solution: a. The equilibrium solutions are \(y_0 = 0\) and \(y_0 = 2\). b. In the direction field, the slope is positive when \(0 < y < 2\) and negative when \(y > 2\). The horizontal lines at \(y = 0\) and \(y = 2\) have constant slopes because they are equilibrium solutions. c. The solution curve for the initial condition \(y(0)=1\) follows the direction field, and as \(t\) increases, it approaches the equilibrium solution \(y = 2\) asymptotically.

Step-by-step solution

Finding equilibrium solutions

To find the equilibrium solutions, set the right side of the differential equation equal to zero: $$y(2-y) = 0$$ This equation factors into: $$y(2-y) = y(2 - y) = 0$$ So, the equilibrium solutions are \(y_0 = 0\) and \(y_0 = 2\).

Sketching the direction field

To sketch the direction field, we must determine the slope at various points \((t, y)\) in the \(t \ge 0\) region. Since the right-hand side of the differential equation is \(y(2-y)\), the slope at any point \((t, y)\) is given by: $$y'(t) = y(2-y)$$ Note that the slope only depends on \(y\) and not on \(t\). In this problem, the slope is positive when \(0 < y < 2\) and negative when \(y > 2\). On the equilibrium solutions \(y_0 = 0\) and \(y_0 = 2\), the slope is zero. Therefore, the horizontal lines at \(y = 0\) and \(y = 2\) have constant slopes. Now, sketch the direction field on a graph with the horizontal axis representing the time \(t\) (since \(t \ge 0\)) and the vertical axis representing \(y\). The direction field consists of small line segments tangent to the solution at various points. For example, the line segments along the horizontal lines \(y = 0\) and \(y = 2\) will be horizontal because the slope is zero. Also, the line segments in the region \(0 < y < 2\) should be positive, while those in the region \(y > 2\) should be negative.

Sketch the solution curve for the initial condition \(y(0) = 1\)

To sketch the solution curve for the initial condition \(y(0) = 1\), note that at \(t = 0\), \(y = 1\). This point lies in the region \(0 < y < 2\) where the slope is positive. So, the solution curve will have a positive slope initially and will follow the direction field. Since the slope decreases as \(y\) approaches the equilibrium solution \(y = 2\), the solution curve will approach \(y = 2\) as \(t\) increases. The solution curve never reaches equilibrium solutions but approaches it asymptotically. In summary, we have found the equilibrium solutions \(y_0 = 0\) and \(y_0 = 2\), sketched the direction field, and sketched the solution curve for the initial condition \(y(0)=1\). The solution curve reveals that the function approaches the equilibrium solution \(y = 2\) as time increases.

Key concepts

Differential Equation

A differential equation is a mathematical equation that relates some function with its derivatives. In the context of the given exercise, we're dealing with an autonomous type of differential equation, which is identified by the form
\( y'(t) = f(y) \).
The term autonomous means that the rate of change of the function \( y(t) \) depends only on its current value and not explicitly on the independent variable, which in this case is time \( t \).

Understanding differential equations is crucial because they model a multitude of phenomena in various fields such as physics, engineering, and biology. For example, they are used to model population growth, motion of particles, and changes in investment over time. The particular equation in our exercise, \( y'(t) = y(2 - y) \), models a situation where the rate of change of a quantity is proportional to both the quantity itself and the difference between the quantity and some maximum value.

Direction Field

Direction fields, also known as slope fields, are visual representations that help us understand the behavior of solutions to a differential equation without actually solving the equation. Each line segment, or arrow, in a direction field indicates the slope of the solution curve at that point.

In constructing a direction field, one typically evaluates the derivative \( y'(t) \) at several points \( (t, y) \) and draws a line segment with that slope at each point. The exercise showcases an autonomous differential equation, meaning the direction field is time-invariant; the slopes depend solely on the value of \( y \), contributing to a repeating pattern vertically along the \( t \) axis.

Direction fields make understanding the qualitative behavior of the solution straightforward. For the given equation, the direction field helps us infer that for \( y \)
between 0 and 2, the solution grows, whereas for \( y \)> 2, the solution decreases. This knowledge is helpful for predicting trends in the absence of an explicit formula for the solution.

Autonomous Equation

An autonomous equation is a type of differential equation where the rate of change of the dependent variable only depends on the dependent variable itself and not on the independent variable. In our exercise, the given autonomous equation is \( y'(t) = f(y) \)
, indicating that the slope of the tangent line to the solution curve at any point is dependent only on \( y \) and not directly on \( t \).

Autonomous equations often come into play when the process being modeled is self-governing and the rate of change is not directly dictated by time but by the state of the system. A key feature of these equations is that their equilibrium solutions are constant solutions where \( f(y_0) = 0 \). These solutions correspond to horizontal lines in the direction field. For \( y'(t) = y(2 - y) \), we have two equilibrium solutions, \( y_0 = 0 \) and \( y_0 = 2 \), meaning these are the values where the rate of change of \( y \) is zero.

Equilibrium solutions are significant because they indicate conditions under which the system remains unchanged, helping us understand the long-term behavior of the system, like whether populations stabilize or investments cease to grow.