Chapter 7.3, Problem 1QQ

To determine

To find which of the following differential equations would produce

Slope field.

Answer to Problem 1QQ

  $\text{Only choice E gave a positive slope so it is the correct answer}\text{.}$

Explanation of Solution

Given information:

  $\begin{array}{l}\left(\text{A}\right)\frac{dy}{dx}=y-3x\left(\text{B}\right)\frac{dy}{dx}=y-\frac{x}{3}\\ \left(C\right)\frac{dy}{dx}=y+\frac{x}{3}\left(\text{D}\right)\frac{dy}{dx}=x+\frac{y}{3}\\ \left(\text{E}\right)\frac{dy}{dx}=x-\frac{y}{3}\end{array}$

  

Calculation : The required solution is obtained as,

  $\begin{array}{l}\text{The slope field only has positive slopes in the fourth quadrant so we can test }\\ \text{a point from the fourth quadrant into each differential equation to eliminate}\\ \text{the answer choices}\text{. Choices C and D are the same differential equation so }\\ \text{choice D is ignored below}\text{. Testing the point }\left(3,-3\right)\text{ gives: }\\ \\ \text{A}\text{. }\frac{dy}{dx}=y-3x\text{ B}\text{. }\frac{dy}{dx}=y-\frac{x}{3}\\ =-3-3\left(3\right)=-3-\frac{3}{3}\\ =-3-9=-3-1\\ =-12=-4\\ \text{C}\text{. }\frac{dy}{dx}=y+\frac{x}{3}\text{ E}\text{. }\frac{dy}{dx}=x-\frac{y}{3}\\ =-3+\frac{3}{3}=3-\frac{-3}{3}\\ =-3+1=3+1\\ =-2=4\\ \\ \text{Hence, only choice E gave a positive slope so it is the correct answer}\text{.}\end{array}$