Chapter 7.1, Problem 50E

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 second quadrant.

Answer to Problem 50E

  

The correct graph is option (b)

Explanation of Solution

Given information:

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

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

Graph:

  

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

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

  $\begin{array}{l}\frac{dy}{dx}=y^2-x\\ \frac{dy}{dx}=1^2-2\\ \frac{dy}{dx}=1-2=-1\end{array}$

Interpretation:

The correct graph is option (b) since it has a line segment with a negative slope at the point. The reflected point in the second quadrant is at $\left(-2,1\right)$ .

Substituting this into the differential equation gives

  $\begin{array}{l}\frac{dy}{dx}=y^2-x\\ \frac{dy}{dx}=1^2-\left(-2\right)\\ \frac{dy}{dx}=1+2=3\end{array}$

Hence, the line segment is drawn through $\left(-2,1\right)$ with a slope of 3.

Therefore, the correct graph is option (b).