Chapter 7.1, Problem 42E

To determine

To match the slope $\frac{dy}{dx}=\sqrt{y^2-4y+5}$ with the correct graph and to find the particular solution for the point $\left(0,2\right)$ .

Answer to Problem 42E

The correct graph of the given slope $\frac{dy}{dx}=\sqrt{y^2-4y+5}$ is,

  

The graph of the point $\left(0,2\right)$ on the given slope is,

  

Explanation of Solution

Given information :

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

Calculation:

Consider, the slope $\frac{dy}{dx}=\sqrt{y^2-4y+5}$ .

Using the above value it is clear that, the slope is in square root which can never be in negative.

Thus, the graphs A is eliminated since they have negative slope.

Consider,

  $\begin{array}{l}\frac{dy}{dx}=0\\ \Rightarrow \sqrt{y^2-4y+5}=0\Rightarrow y^2-4y+5=0\end{array}$

Thus, $y^2-4y+5=0$ has no solution and hence the slope will never equals to zero.

This ruins out the graphs B, C and E snice these graphs have slopes of Zero.

Therefore, the graph of D or F is the match for the slope $\frac{dy}{dx}=\sqrt{y^2-4y+5}$ .

Since $\frac{dy}{dx}$ is in terms of y only so all the points with the same y- coordinate must have the same slope.

This eliminates the graph D since all the points in the graph have the same slopes when the x -coordinates are the same.

Hence, the graph of the slope field is in the form,

  

Thus, from the given set of graphs option F is the correct match for the slope $\frac{dy}{dx}=\sqrt{y^2-4y+5}$

Now, to find the particular solution through the above graph for the point $\left(0,2\right)$ .

Consider the point $\left(0,2\right)$ by using the line segment at the point $x=1$ sketch the additional portion of the curve to have the slope and similarly using the line segment at the point $x=0$ again sketch the curve continuing the similar process to fill the entire sketch then the final curve results as,

  

Which is the particular solution for the slope $\frac{dy}{dx}=\sqrt{y^2-4y+5}$ at the point $\left(0,2\right)$ .