Question

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 \(y(0)=1\). $$y^{\prime}(t)=y(y-3)$$

Short answer

Answer: The equilibrium solutions are \(y = 0\) and \(y = 3\). The solution curve for the initial condition \(y(0) = 1\) starts at \((0,1)\) and follows the direction field without touching the equilibrium lines.

Step-by-step solution

Find the equilibrium solutions

To find the equilibrium solutions, we will set \(y^{\prime}(t)=0\) and solve for \(y\): $$y(y-3) = 0$$ This equation is true when either \(y = 0\) or \(y = 3\). Therefore, the equilibrium solutions are \(y = 0\) and \(y = 3\).

Sketch the direction field

To sketch the direction field of the given differential equation \(y^{\prime}(t) = y(y-3)\), we need to compute the derivative for a range of \(y\) values and plot the corresponding slopes. Following the slope of the direction field, we can visualize how the function behaves: - For \(y < 0\), we have \(y^{\prime}(t) > 0\) (positive slope). - For \(0 < y < 3\), we have \(y^{\prime}(t) < 0\) (negative slope). - For \(y > 3\), we have \(y^{\prime}(t) > 0\) (positive slope). Now we can sketch the direction field, keeping in mind that equilibrium solutions are horizontal lines (constant \(y\) values), and the slopes vary about the \(y\) values.

Sketch the solution curve for given initial condition

Now let's sketch the solution curve for the initial condition \(y(0) = 1\). Locate the point \((0, 1)\) on the \(ty\)-plane, and note that it is between the equilibrium solutions found previously (\(y = 0\) and \(y = 3\)). Following along the direction field from the initial condition \((0,1)\), we can see that the solution curve will pass through points with negative slopes on the \(ty\)-plane. The curve will begin at \((0,1)\), following the direction of the negative slope, without touching the equilibrium lines (as those are constant solutions). In conclusion: - The equilibrium solutions are \(y = 0\) and \(y = 3\). - The direction field can be sketched using the slopes computed depending on \(y\). - The solution curve for the initial condition \(y(0) = 1\) starts at \((0,1)\) and follows the direction field without touching the equilibrium lines.

Key concepts

Equilibrium Solutions

An equilibrium solution for a differential equation represents a state where the system remains unchanging. For an autonomous differential equation expressed as \( y'(t) = f(y) \), an equilibrium solution occurs at constant values of \( y \) where the derivative equals zero. This implies the system is in balance.

To find these points, we solve \( f(y) = 0 \). For the given problem \( y'(t) = y(y - 3) \), setting this equal to zero gives us the points \( y = 0 \) and \( y = 3 \). At these points, \( y'(t) = 0 \), indicating no change in \( y \) over time. These are the constant solutions where the differential equation remains static.

Understanding equilibrium solutions is crucial as they often represent stable states in a system, like an object's rest position under the influence of forces. In a graph, these solutions are represented by horizontal lines, which help in visualizing the behavior of the system at various points.

Direction Field

A direction field, sometimes called a slope field, is a visual tool used to represent the behavior of solutions to a differential equation. For an autonomous differential equation like \( y'(t) = y(y-3) \), the direction field shows how the slope of solutions changes with respect to \( y \) at each point.

To construct a direction field, we first determine the slope \( y'(t) \) at various values of \( y \):

• For \( y < 0 \), the slope is positive, implying that the solution curves rise as \( t \) increases.
• For \( 0 < y < 3 \), the slope is negative, so the solution curves fall as \( t \) increases.
• For \( y > 3 \), the slope becomes positive again, indicating rising curves.

These patterns help us sketch the direction field by drawing short line segments on the \( ty \)-plane indicating these slopes. Equilibrium solutions appear as horizontal lines in this field, marking points where \( y'(t) = 0 \), thus no slope or change.

Using a direction field allows us to get an intuitive grasp of how solutions behave over time and aids in predicting long-term behavior without solving the equation explicitly.

Solution Curve

A solution curve represents a specific solution to a differential equation given an initial condition. In this context, the curve is traced in the \(ty\)-plane starting from an initial condition, such as \( y(0) = 1 \).

For the differential equation \( y'(t) = y(y-3) \), we start at the point \( (0,1) \) on the \(ty\)-plane. This point falls between the equilibrium solutions \( y = 0 \) and \( y = 3 \), where the slope \( y'(t) \) is negative. This indicates that the solution curve will tend downwards, reflecting a decrease in \( y \) over time.

As we follow the direction field from \( (0,1) \), the curve follows the path dictated by the slope, moving in a direction determined by the interplay of forces described by the differential equation. Solution curves are crucial for visualizing and understanding the behavior of a system under given conditions, providing insights into how variables evolve over time relative to starting conditions.