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.