Question

Draw a direction field for the given differential equation and state whether you think that the solutions are converging or diverging. $$ y^{\prime}=y(3-t y) $$

Short answer

Explain your reasoning.

Step-by-step solution

Understand the given differential equation

The given differential equation is: $$ y^{\prime}=y(3-t y) $$ This is a first-order, nonlinear equation where the derivative of the dependent variable, y, with respect to the independent variable, t, depends on the product of y and \((3 - ty)\).

Draw a grid of points in the t-y plane

We will draw a grid of points in the t-y plane, with t on the horizontal axis and y on the vertical axis. Choose an appropriate range for t and y, such as, t going from -3 to 3 and y going from -3 to 3.

Find the slope at each grid point

With the grid points in place, we will determine the slope of the solution curve passing through each point. To do this, we'll use the equation for the derivative: $$ y^{\prime}=y(3-t y) $$ Find the slope by plugging in the grid point coordinates (t, y) and computing the value of the derivative (the slope) at that point.

Draw segments with these slopes at each grid point

Now that we have the slope at each grid point, we'll draw small line segments with these slopes at each grid point on the t-y plane. These short line segments will represent the direction field.

Examine the direction field to determine whether the solutions are converging or diverging

Observe the direction field drawn in the previous step. If the line segments appear to be converging toward a particular point or curve, we can conclude that the solutions of the given differential equation are converging. On the other hand, if the line segments appear to be diverging away from a certain point or curve, we can conclude that the solutions are diverging. Sometimes, the situation may not be clearly converging or diverging; in that case, we can describe the behavior of the solutions qualitatively or look for underlying patterns. In conclusion, follow these steps to analyze the given differential equation, draw its direction field, and determine whether the solutions are converging or diverging.

Key concepts

Differential Equations

Differential equations are a fundamental part of mathematics that describe how a particular quantity changes over time. In our exercise, the equation \( y' = y(3 - ty) \) represents a first-order, nonlinear differential equation. Here, \( y' \) is the derivative of \( y \) with respect to \( t \), which tells us the rate at which \( y \) changes as \( t \) changes. This equation is nonlinear due to the term \( ty \), which involves a product of variables. Analyzing such equations often involves finding a solution that explains how \( y \) behaves over time with initial conditions. Differential equations are widely used in fields like physics, engineering, and biology to model real-world phenomena, from population growth to the motion of objects.

Convergence and Divergence

In the context of differential equations, convergence and divergence describe the behavior of solution curves over time. A solution is said to be converging if it approaches a fixed value or curve as \( t \) increases. Conversely, a solution is diverging if it moves away from a point or a set trajectory. For the differential equation \( y' = y(3 - ty) \), observing a direction field can help us quickly see the likely behavior of solutions. If the line segments in the direction field move closer together and point to a common area, the solutions are converging. If they spread apart, the solutions are diverging. Understanding these concepts can help predict long-term behavior of dynamic systems.

Nonlinear Equations

Nonlinear equations, such as \( y' = y(3 - ty) \), involve terms that are not just simple multiples of the variables. The presence of the term \( ty \) in this equation makes it nonlinear. Nonlinear equations can exhibit complex behaviors such as limit cycles, chaos, or multiple solutions, depending on initial conditions and parameters. These equations are more challenging to solve analytically compared to linear equations and often require numerical methods for exploration. The nonlinearity can lead to unique patterns in the direction fields, reflecting the diverse possible trajectories of the solutions.

Slope Fields

Slope fields, also known as direction fields, are visual tools that help us understand differential equations without solving them explicitly. For an equation like \( y' = y(3 - ty) \), we use a grid in the \( t-y \) plane and compute the slopes given by the equation at various points. By plotting small line segments at each point with these slopes, we create a direction field. These line segments collectively demonstrate the behavior of the solutions to the differential equation. Slope fields can reveal whether solutions tend to converge to a stable value or diverge away from certain areas. This visual representation is a powerful method for gaining insight into complex equations without delving into intricate calculations.