Chapter 7, Problem 38RE

To determine

To construct the slope field for given differential equation by slope analysis not using the graphing calculator.

Explanation of Solution

Given information :

A graph with lattice point is shown below. Given differential equation is $\frac{dy}{dx}=1-y$

As per the given lattice points as shown in above graph one can obtain the below points

  $\begin{array}{l}\left(0,1\right),\left(-1,0\right),\left(1,0\right),\left(-1,1\right),\left(-1,2\right),\left(-1,-1\right),\left(-1,2\right),\left(-1,-1\right),\\ \left(1,2\right),\left(1,-1\right),\left(0,1\right),\left(0,2\right)\text{ and }\left(0,-1\right)\end{array}$

At these points find out the slope using the given differential equation

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

Interpretation :

Now using these points draw the slope field according to the slope obtained at the lattice point with tiny segments.

Hence the reciprocal slope field is obtained as above.