Chapter 6.1, Problem 40E

To determine

To choose : The correct graph.

Answer to Problem 40E

The correct graph is: Graph $(f)$

  

Explanation of Solution

Given information:

The differential equation is: $\frac{dy}{dx}=-\frac{x}{y}$

Calculation:

For $\frac{dy}{dx}=-\frac{x}{y}$ , when

  $x=0$ and

  $y\ne 0$

The slopes will be:

  $\begin{array}{c}\frac{dy}{dx}=-\frac{y}{x}\\ =-\frac{0}{x}\\ =0\end{array}$

Graphs $(a)$ , $(c)$ , (d), and $(e)$ can then be eliminated since they don't have slopes of 0 when $x=0$ . The correct answer is then either graph (c) or $(f)$ . Consider the point $(3,2)$ .

The slope must be

  $\begin{array}{c}\frac{dy}{dx}=-\frac{x}{y}\\ =-\frac{3}{2}<0.\end{array}$

Graph $(c)$ has positive slopes around $(3,2)$ while graph $(f)$ has negative slopes. The correct graph is then graph $(f)$ .

Sketch a section of the curve with a roughly equal slope using the line segment at that location. Sketch another section of the curve with the same slope as the line segment next to  sketch at $x=2$ . Until filled in the entire sketch, keep doing this on both sides of the $x=3$ equation. The drawing is displayed below.

  

Therefore the required graph $(f)$ is match the given differential equation with the appropriate slope field.