Chapter 2: Q16E (page 45)
(a) Consider the direction field of the differential equation but do not use technology to obtain it. Describe the slopes of the linear elements on the lines x=0, y= 3,y =4 and y = 5.
(b) Consider the IVP, y(0)= y0, where y0< 4. Can a solution
such as
? Based on the information in part (a), discuss.
Short Answer
Answer:
a) At the lineand
, the slope of the elements in the direction field is always equal to
, and along the line
and
, the slope is given by
.
b) No, it cannot pass this line and such a solution cannot tend toas
.
Step by step solution
Direction field.
If we systematically evaluateover a rectangular grid of points in the
-plane and draw a line element at each point
of the grid with a slope
then the collection of all these line elements is called a direction field or a slope field of the differential equation
.
Solution for part a).
Whenwe have
.
Therefore, along the line, the slope of the elements in the direction field is always equal to
.
Whenwe have
. Therefore, along the line
, the slope is given by
. For instance, at
the slope is zero and at
, the slope is
.
Whenwe have
. Therefore, along the line
, the slope is equal to
.
Whenwe have
. Therefore, along the line
, the slope is given by
. For instance, at
the slope is zero and at
, the slope is
.
Solution for part b).
We know that along the linewe always have a negative slope. Suppose a solution curve starts somewhere below
on the y-axis and goes to
as
.
But at the line, this curve must be decreasing, not increasing. Therefore it cannot pass this line and such a solution cannot tend to
as
.
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