Chapter 7.1, Problem 49E

To determine

To graph:

To graph an only possible graph and draw a line for the same slope field through the reflected point in the first quadrant.

Answer to Problem 49E

  

The correct graph is option (c)

Explanation of Solution

Given information:

The point in the second quadrant is $\left(-2,1\right)$ .

A single line from the slope field is $\frac{dy}{dx}=2y+x$

Graph:

  

The point in the second quadrant is $\left(-2,1\right)$ .

Now substituting this point into the differential equation $\frac{dy}{dx}=2y+x$

  $\begin{array}{l}\frac{dy}{dx}=2y+x\\ \frac{dy}{dx}=2\left(1\right)+\left(-2\right)\\ \frac{dy}{dx}=2-2=0\end{array}$

Interpretation:

The correct graph is option (c) since it has a horizontal line segment at the point. The reflected point in the first quadrant is at $\left(2,1\right)$ .

Substituting this into the differential equation gives

  $\begin{array}{l}\frac{dy}{dx}=2y+x\\ \frac{dy}{dx}=2\left(1\right)+\left(2\right)\\ \frac{dy}{dx}=2+2=4\end{array}$

Hence, the line segment is drawn through $\left(2,1\right)$ with a slope of 4.

Therefore, the correct graph is option (c).