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. $$\frac{d y}{d x}=-\frac{y}{x}$$

Short answer

The key to visualizing the solution of a differential equation is to draw little line segments at the given points that represent the slope of the solution at that point. For the differential equation \(\frac{d y}{d x}=-\frac{y}{x}\), calculate the slopes at several points on the plane, then create a matching line segment at each point. Please note that this short summary does not include the slope field graph. It is also important to note that the actual values of \(x\) and \(y\) can change, which would change the slopes and in turn change the slope field.

Step-by-step solution

Understanding Slope Fields

The slope field of a differential equation \(y' = f(x, y)\) is a graph that at the point \((x,y)\), shows the slope of the solution to the differential equation that passes through that point. For every point in the plane, plug it into the differential equation to get the slope of the solution at that point. Draw a short line segment with that slope at the point.

Evaluate Slopes

Now, we have to find the slope \(m\) at several points. It's the value of the derivative at that point, which, in our case, is \(\frac{d y}{d x}=-\frac{y}{x}\). Suppose we choose the points \((2,2), (2,-2), (-2,2)\) and \((-2,-2)\). Plugging these values in the given differential equation, we will get the associated slopes. For \((2,2)\), \(m = -\frac{y}{x} = -\frac{2}{2} = -1\). Similar calculations provide \(m = 1\) at \((2,-2)\), \(m = 1\) at \((-2,2)\) and \(m = -1\) at \((-2,-2)\).

Draw the Slope Field

Draw a small line segment at each point with the slope you calculated. Make sure that the direction and the steepness of all these line segments correspond correctly to the values of the slope. Each tiny line segment gives the direction of the solution curve passing through it at the point. As more segments are added, the shape of the solution curve becomes apparent.