   Chapter 6.1, Problem 65E

Chapter
Section
Textbook Problem

Slope Field Use the slope field for the differential equation y ′ = 1 / x , where x > 0 to sketch thegraph of the solution that satisfies each given initial condition. Then make a conjecture about thebehavior of a particular solution of y ′ = 1 / x as x → ∞ . To print an enlarged copy of the graph, goto MathGraphs.com. a). ( 1 , 0 ) b). ( 2 , − 1 )

a)

To determine

To Graph: The solution that satisfies the given initial condition and then make a conjecture about the behavior of a particular solution of y=1/x as x.

Explanation

Given: The differential equation y=1/x, where x>0 and the point (1,0). The slope field for the differential equationis represented by the graph.

Graph:

We start by creating a table showing the slopes at several points.

The table shown below is a smallsample.

The slopes at many other points should be calculated to get a representative slope field.

 x 0 0 1 1 2 2 3 3 4 5 y -1 1 -1 1 -1 1 -1 1 -1 1 y′=1x 1 1 0.5 0.5 0.33 0.33 0.25 0.2

Now, we draw line segments at the points with their respective slopes as shown below.

After the slope field is drawn, we start from the starting point and the line goes right towards the direction of the line segment

b)

To determine

To Graph: The solution that satisfies the given initial condition and then make a conjecture about the behavior of a particular solution of y=1/x as x.

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