Chapter 7.1, Problem 44E

To determine

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

Answer to Problem 44E

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

  

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

  

Explanation of Solution

Given information :

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

Calculation:

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

Using the above value it is clear that, when $y=x$ the slope become zero.

Hence the slope field must have horizontal line segments for all points on $y=x$ .

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

  

Therefore, from the given set of graphs option E is the correct match for the slope $\frac{dy}{dx}=\left|x-y\right|$

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 $\frac{dy}{dx}=\left|x-y\right|$ at the point $\left(0,2\right)$ .