Question

In Exercises \(29-34\) , construct a slope field for the differential equation. In each case, copy the graph at the right and draw tiny segments through the twelve lattice points shown in the graph. Use slope analysis, not your graphing calculator. $$\frac{d y}{d x}=-\frac{x}{y}$$

Short answer

The slope field can be constructed for the differential equation \( \frac{d y}{d x} = -\frac{x}{y} \) by calculating the slope at 12 grid points from the equation and drawing a tiny line segment at that point with that slope. The slope field obtained will provide a graphical representation of the solutions to the differential equation.

Step-by-step solution

Understand the equation

The given equation is a differential equation: \( \frac{d y}{d x} = -\frac{x}{y} \). This means that the slope of any solution curve at any point (x,y) is given by -x/y.

Choose grid points

As per the problem instructions, we are going to draw tiny line segments at 12 different points in the graph. Assume the graph to be on a grid of equally spaced horizontal and vertical lines (i.e. a 'lattice'). Choose 12 points on this lattice for which to calculate the slopes.

Calculate slopes and draw segments

Substitute the x and y coordinate of each point into \( \frac{d y}{d x} = -\frac{x}{y} \) equation to calculate the slope. Then, draw a tiny line segment at that point that has that slope. The slope field is constructed by repeating this process for each of the 12 points.

Interpret the slope field

Now that the slope field has been constructed, try to interpret what it represents in terms of the differential equation. Any curve drawn through the graph must have at each point the same slope as indicated by short line segments in the slope field. This means that the slope field gives a good idea of the general shape and direction of solution curves.