Question

In Exercises \(12-22\) construct a direction field and plot some integral curves in the indicated rectangular region. $$ y^{\prime}=2-3 x y ; \quad\\{-1 \leq x \leq 4,-4 \leq y \leq 4\\} $$

Short answer

Answer: The main purpose of creating a direction field and plotting integral curves for a given differential equation is to visually represent and analyze the behavior of the solutions in a specified rectangular region. This helps in better understanding the direction and rate of change of the function at different points.

Step-by-step solution

Create a grid in the specified rectangular region

In the specified rectangular region (-1 ≤ x ≤ 4, -4 ≤ y ≤ 4), create a grid with equally spaced points. You can manually do this or use graphing software for accuracy and better visualization.

Calculate the slope of the tangent line at each point on the grid

For each point (x, y) on the grid, calculate the slope of the tangent line using the given differential equation: y' = 2 - 3xy For example, if a point on the grid is (0, 0), the slope of the tangent line at this point would be: y'(0, 0) = 2 - 3(0)(0) = 2 Repeat this process for each point on the grid.

Draw short line segments representing the tangent lines at each point on the grid

Now that you have the slope of the tangent line at each point, draw a short line segment representing the tangent line at each grid point. The direction and steepness of the tangent lines indicate the direction and rate of change of the function at that point.

Plot some integral curves by following the tangent lines

Starting from some initial conditions (x0, y0) in the rectangular region, follow the tangent lines to plot an integral curve, also known as a solution curve. The integral curve is an approximation of the solution to the given differential equation. You can plot multiple integral curves for different initial conditions within the specified region.

Observe the behavior of the solution

By looking at the direction field and the integral curves plotted, observe and analyze the behavior of the solutions to the given differential equation. This visual representation will help you better understand the behavior of the solutions in the specified rectangular region.

Key concepts

Differential Equations

A differential equation is a type of equation that relates a function with its derivatives. In essence, it describes how a particular quantity changes over time or space. For the differential equation given in the exercise, y' = 2 - 3xy, the rate at which y changes with respect to x is governed by the function's current values of x and y.

Understanding differential equations is vital because they are used to model real-world phenomena in fields such as physics, engineering, economics, and biology. Solving a differential equation means finding a function (or set of functions) that satisfies the relationship outlined by the equation. The difficulty of solving a differential equation can vary greatly, from simple separable equations to complex non-linear systems that cannot be solved analytically and require numerical methods.

Integral Curves

An integral curve, also known as a solution curve, is a graphical representation of the solution to a differential equation. It is a curve in the domain of the independent variable, which in this case is x, that maps out the trajectory of the dependent variable, y, based on the differential equation. The exercise involves plotting these integral curves by following the tangent lines that represent the derivative at each point.

In the rectangular region given, these curves can help visualize the solution and understand how initial values affect the behavior of the system. Integral curves are especially useful when an exact analytic solution to a differential equation is not available and we rely on qualitative analysis or numerical approximation methods to study the system's behavior.

Tangent Line Slope

The slope of a tangent line to a curve at a given point represents the instantaneous rate of change of the curve at that point. For a function y = f(x), the derivative f'(x) gives the slope of the tangent line at any point x. This concept is crucial when working with differential equations because the differential equation itself essentially provides a formula for the slope of the tangent line at any point on an integral curve.

In practice, by calculating the slope of the tangent line at several points, as you would in Step 2 of the exercise, and then drawing these lines on the graph, you create a so-called direction field. This field gives a visual sense of how the function behaves in the region of interest. The pattern can reveal equilibrium points, where the slope is zero, and areas of stability or instability within the solution space.