Chapter 7.1, Problem 39E

To determine

To match the slope $\frac{dy}{dx}=-\frac{y}{x}$ with the correct graph and to find the particular solution for the point $\left(3,2\right)$ .

Answer to Problem 39E

The correct graph of the given slope $\frac{dy}{dx}=-\frac{y}{x}$ is,

  

The graph of the point (3,2) on the given slope is,

  

Explanation of Solution

Given information :

The slope $\frac{dy}{dx}=-\frac{y}{x}$ .

Calculation:

Consider, the slope $\frac{dy}{dx}=-\frac{y}{x}$ .

Using the above value it is clear that, when $y=0\text{and }x\ne 0$ the slope is $\frac{dy}{dx}=-\frac{y}{x}=-\frac{0}{x}=0$ .

Hence the slope can only be zero when $y=0$ .

Thus, the graphs A, C, D and F can be eliminated since they don’t have slopes of zero when $y=0$ .

Therefore, the graph of B or E is the match for the slope $\frac{dy}{dx}=-\frac{y}{x}$ .

Consider the point $\left(3,2\right)$ substitute it in the slope $\frac{dy}{dx}=-\frac{y}{x}$ ,

  $\Rightarrow \frac{dy}{dx}=-\frac{2}{3}$

(i.e.,) The slope is negative at the point $\left(3,2\right)$ . But on the graph of Ehave positive slopes at the point $\left(3,2\right)$ .

Hence, the graph of the slope field is in the form,

  

Thus, from the given set of graphs option B is the correct match for the slope $\frac{dy}{dx}=-\frac{y}{x}$ .

Now, to find the particular solution through the above graph for the point $\left(3,2\right)$ .

Consider the point $\left(3,2\right)$ by using the line segment at the point $x=2$ sketch the additional portion of the curve to have the slope and similarly using the line segment at the point $x=3$ again sketch the curve continuing the similar process to fill the entire sketch then the final curve results as,

  

Which is the particular solution for the slope $\frac{dy}{dx}=-\frac{y}{x}$ at the point $\left(3,2\right)$ .