Chapter 7.1, Problem 46E

To determine

To match the slope $\frac{dy}{dx}=xy$ with the correct graph and to find the particular solution for the point $\left(0,2\right)$ .

Answer to Problem 46E

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

  

The graph of the point $\left(0,2\right)$ on the given slope is,

  

Explanation of Solution

Given information :

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

Calculation:

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

It is clear that the slope is zero whenever $x=0$ or $y=0$ .

This shows that the graph must have horizontal line segments at all points with an x-coordinate of zero and all points with a y-coordinates of zero.

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

  

Therefore, from the given set of graphs option A is the correct match for the slope $\frac{dy}{dx}=xy$

Now, to find the particular solution through the above graph for the point $\left(0,2\right)$ .

Consider the point $\left(0,2\right)$ by using the line segment at the point $x=1$ sketch the additional portion of the curve to have the slope and similarly using the line segment at the point $x=0$ 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 $\frac{dy}{dx}=xy$ at the point $\left(0,2\right)$ .