Answered: y A - /||| I - /|| //||| ////. ||||||…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Identifying direction fields Which of the differential equations
a–d corresponds to the following direction field? Explain your reasoning.
a. y '(t) = 0.5(y + 1)(t - 1)
b. y '(t) = -0.5(y + 1)(t - 1)
c. y '(t) = 0.5(y - 1)(t + 1)
d. y '(t) = -0.5(y - 1)(t + 1)

y A
- /||| I
- /||
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Expert Solution

Step 1: Given:

Direction field:

To determine which of the differential equations corresponds to the following direction field:

$table attributes columnalign right center left columnspacing 0px end attributes row cell a. space y space apostrophe left parenthesis t right parenthesis space end cell equals cell space 0.5 left parenthesis y space plus space 1 right parenthesis left parenthesis t space minus space 1 right parenthesis end cell row cell b. space y space apostrophe left parenthesis t right parenthesis space end cell equals cell space minus 0.5 left parenthesis y space plus space 1 right parenthesis left parenthesis t space minus space 1 right parenthesis space end cell row cell c. space y space apostrophe left parenthesis t right parenthesis space end cell equals cell space 0.5 left parenthesis y space minus space 1 right parenthesis left parenthesis t space plus space 1 right parenthesis space end cell row cell d. space y space apostrophe left parenthesis t right parenthesis space end cell equals cell space minus 0.5 left parenthesis y space minus space 1 right parenthesis left parenthesis t space plus space 1 right parenthesis end cell end table$

Step 2: Explanation:

From the direction field, it can be observed that the slope is zero at $y equals negative 1$ and $t equals 1$ .

Thus, the equation will be of the form: $y apostrophe left parenthesis t right parenthesis equals a left parenthesis y plus 1 right parenthesis left parenthesis t minus 1 right parenthesis$ .

Hence, c and d cannot be the required option.

Now, for $y greater than negative 1$ and $t greater than 1$

$slope space left parenthesis y apostrophe right parenthesis space greater than 0 space$

Thus, $a greater than 0$

Hence, the correct choice is a. $y space apostrophe left parenthesis t right parenthesis space equals space 0.5 left parenthesis y space plus space 1 right parenthesis left parenthesis t space minus space 1 right parenthesis$

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