Chapter 7.1, Problem 34E

To determine

To analyze the slope $\frac{dy}{dx}=x-2y$ .

Answer to Problem 34E

The slope field of the given slope $\frac{dy}{dx}=x-2y$ is,

  

Explanation of Solution

Given information :

The slope $\frac{dy}{dx}=x-2y$ and the points,

  

Calculation:

Consider, the slope $\frac{dy}{dx}=x-2y$ .

The set of points mentioned in the given field is,

  $\left(-1,1\right),\left(-1,0\right),\left(-1,1\right),\left(-1,2\right),\left(0,1\right),\left(0,0\right),\left(0,1\right),\left(0,2\right),\left(1,1\right),\left(1,0\right),\left(1,1\right),\left(1,2\right)$ .

Substituting the points in the given slope,

  $\begin{array}{l}\frac{dy}{d{x}_{\left(-1,2\right)}}=\left(-1\right)-2\left(2\right)=-1-4=-5\\ \frac{dy}{d{x}_{\left(-1,1\right)}}=\left(-1\right)-2\left(1\right)=-1-2=-3\\ \frac{dy}{d{x}_{\left(-1,0\right)}}=\left(-1\right)-2\left(0\right)=-1-0=-1\\ \frac{dy}{d{x}_{\left(-1,-1\right)}}=\left(-1\right)-2\left(-1\right)=-1+4=3\\ \frac{dy}{d{x}_{\left(0,2\right)}}=\left(0\right)-2\left(2\right)=0-4=-4\\ \frac{dy}{d{x}_{\left(0,1\right)}}=\left(0\right)-2\left(1\right)=0-2=-2\\ \frac{dy}{d{x}_{\left(0,0\right)}}=\left(0\right)-2\left(0\right)=0-0=0\\ \frac{dy}{d{x}_{\left(0,-1\right)}}=\left(0\right)-2\left(-1\right)=0+4=4\end{array}$

  $\begin{array}{l}\frac{dy}{d{x}_{\left(1,2\right)}}=\left(1\right)-2\left(2\right)=1-4=-3\\ \frac{dy}{d{x}_{\left(1,1\right)}}=\left(1\right)-2\left(1\right)=1-2=-1\\ \frac{dy}{d{x}_{\left(1,0\right)}}=\left(1\right)-2\left(0\right)=1-0=1\\ \frac{dy}{d{x}_{\left(1,-1\right)}}=\left(1\right)-2\left(-1\right)=1+4=5\end{array}$

Using the above points the direction fields of the slope can be drawn as,