Chapter 7.1, Problem 36E

To determine

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

Answer to Problem 36E

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

  

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

  

Explanation of Solution

Given information :

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

Calculation:

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

Using the above value it is clear that the slope depends only on the value of y. Thus, all the points are with the same y-coordinate.

(i.e.,) If $y=3$ then $\frac{dy}{dx}=y=3$ ; $y=-3$ then $\frac{dy}{dx}=-3$

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

  

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

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 $\frac{dy}{dx}=y$ at the point $\left(3,2\right)$ .