Chapter 6.1, Problem 42E

To determine

To choose : The correct graph.

Answer to Problem 42E

The correct graph is: Graph $(f)$

  

Explanation of Solution

Given information:

The differential equation is: $\frac{dy}{dx}=\sqrt{y^2-4y+5}$

Calculation:

For $\frac{dy}{dx}=\sqrt{y^2-4y+5}$

The slope can never be negative since it is equal to a square root, which can never be negative. This rules out graph $(a)$ because it contains downward slopes. $\frac{dy}{dx}=0$ if

  $y^2-4y+5=0$

  $y^2-4y+5=0$ has no solution though so $\frac{dy}{dx}$ will never equal $0$ .

Thus, since graphs $(b)$ , $(c)$ , and $(e)$ have slopes of 0, they can be disregarded. Thus, graph $(d)$ or is the appropriate response $(f)$

Sine $\frac{dy}{dx}$ is in terms of $y$ only, all points with the same $y$ -coordinate must have the same slope. For example, all points with an $y$ -coordinate of 1 must have a slope of

  $\begin{array}{c}\frac{dy}{dx}=\sqrt{1^2-4(1)+5}\\ =\sqrt{2}\end{array}$

This rules out graph $(d)$ since its points have the same slopes when the $x$ -coordinates, not the $y$ -coordinates, are the same. The appropriate response is then graph $(f)$ .

Start at $(0,2)$ to begin drawing the graph of the specific answer $(0,2)$ . 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 close to  sketch at $x=1$ Up to the point where the sketch is completely filled in, keep doing this on both sides of $x=0$ . The drawing is displayed below.

  

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