Chapter 6.1, Problem 36E

To determine

To choose : The correct graph.

Answer to Problem 36E

The correct graph is: graph $(e)$

  

Explanation of Solution

Given information:

The differential equation is: $\frac{dy}{dx}=y$

Calculation:

For $\frac{dy}{dx}=y$ , only the $x$ -value will affect how the slopes change. As a result, there must be a point at each and every point with the same $x$ -value. For example, all the points with a $x$ -coordinate of $2$ must have a slope of

  $\begin{array}{c}\frac{dy}{dx}=y\\ =2\end{array}$

The correct graph is then graph $(e)$ . To sketch the graph of the particular solution through the point , $(3,2)$ start at $(3,2)$

Sketch a section of the curve with a roughly equal slope using the line segment at that location. Sketch another section of the curve with the same slope as the line segment next to  sketch at $x=2$ . Until filled in the entire sketch, keep doing this on both sides of the $x=3$ equation. The drawing is displayed below.

  h

Therefore the required graph $(e)$ is match the given differential equation with the appropriate slope field.