   Chapter 6.1, Problem 66E

Chapter
Section
Textbook Problem

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

Explanation

Given: The differential equation y=1/y, where y>0 and the point (0,1). 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 -2 -2 -1 -1 0 0 1 1 2 2 y -1 1 -1 1 -1 1 -1 1 -1 1 y′=1/y -1 1 -1 1 -1 1 -1 1 -1 1

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

On solving the differential equation y=1/y, we have

dydx=1y

ydy=dx

So, Integrating both sides, we get

ydy=(dx)

12y2=x<

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/y as x.

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