Please answer the questions. they are connected. ty! Consider the following differential equation. (A computer algebra system is recommended.)y' + = 7 cos 2t, t> 01/2 t(a) Draw a direction field for the given differential equation.y2.0!!!1.51.00.5y4|||||||||||||||||||||||||1||||||||||||www.800||||||||11111O-412|||||||||||||||||||||||||||||||||4|||||||||||0.5111111||||||||||||||||||||||||||| | | | | | | | | |6 8 101||||||||||||||||||||1111111111111111111111111||||||1 1 1 1 1 1 1 | | |******111111||||||ty4O-41y411||||||||||||||||||||110-4111||||||||||11|||||||||0.5|||||1/ 111III- 1 | | | \ / |||||||-11|| \/ ||VIIIIII11-1116t (b) Based on an inspection of the direction field, describe how solutions behave for large t.O All solutions seem to approach a line in the region where the negative and positive slopes meet each other.O All solutions seem to eventually have positive slopes, and hence increase without bound.O All solutions seem to eventually have negative slopes, and hence decrease without bound.O The solutions appear to be oscillatory.O If y(0) > 0, solutions appear to eventually have positive slopes, and hence increase without bound. If y(0) ≤ 0, solutions appear to have negative slopes and decrease without bound.(c) Find the general solution of the given differential equation.y(t) =Use it to determine how solutions behave as t → co.7O All solutions converge to the function y = cos 2t.2O All solutions will increase exponentially.O All solutions converge to the function y =27O All solutions converge to the function y =2O All solutions will decrease exponentially.sin 2t.

Answered: Image /qna-images/answer/9f89704c-6821-4da8-bf26-eb443325e073.jpg

Consider the following differential equation. (A computer algebra system is recommended.) y' + = 7 cos 2t, t> 0 1/2 t (a) Draw a direction field for the given differential equation. y 2.0!! 1.0 1.5▬▬▬▬▬ 1|||||||||||| …………………………… ||||||| 0.5 ||||||||| O-4 |||||||||||| y 41 2 ||| |||||||||||||||| |||||||||||||||||| 4 ***||| 0.5 6 8 10 11111111 |||||||||||||||| ||||||||||||||| 1111111111111111 1111111111 | | | | | | | | | | | | |||||| ||||||| 111111 t y 4 O-41 y 4 11 |||||||||||||| 11 0-4111 111111 ||||||||||| |||||||||| 0.5 ||||| |||||||| VIII 111 |||||||| 1|||||||| 71||||||ii 1/ 111 III III -1 || | \//| | | | | / \ || -1111 VIL \ || | \/ || | || / \ \ XIII/II/N 11 11-11 16 t

Consider the following differential equation. (A computer algebra system is recommended.) y' + = 7 cos 2t, t> 0 1/2 t (a) Draw a direction field for the given differential equation. y 2.0!! 1.0 1.5▬▬▬▬▬ 1|||||||||||| …………………………… ||||||| 0.5 ||||||||| O-4 |||||||||||| y 41 2 ||| |||||||||||||||| |||||||||||||||||| 4 ***||| 0.5 6 8 10 11111111 |||||||||||||||| ||||||||||||||| 1111111111111111 1111111111 | | | | | | | | | | | | |||||| ||||||| 111111 t y 4 O-41 y 4 11 |||||||||||||| 11 0-4111 111111 ||||||||||| |||||||||| 0.5 ||||| |||||||| VIII 111 |||||||| 1|||||||| 71||||||ii 1/ 111 III III -1 || | \//| | | | | / \ || -1111 VIL \ || | \/ || | || / \ \ XIII/II/N 11 11-11 16 t

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

Please answer the questions. they are connected. ty!

Consider the following differential equation. (A computer algebra system is recommended.)
y' + = 7 cos 2t, t> 0
1/2 t
(a) Draw a direction field for the given differential equation.
y
2.0!!!
1.5
1.0
0.5
y
4
||||||||||||
|||||||||||||
1||||||||||||
www.800
||||||||
11111
O-41
2
|||
||||||||||||||
||||||||||||||||
4
|||||||||||
0.5
111111|||||||||
|||||||||||||||||
| | | | | | | | | |
6 8 10
1||||||||||||||||||||
1111111111111111
111111111||||||
1 1 1 1 1 1 1 | | |
******
111111
||||||
t
y
4
O-41
y
4
11
||||||||||||||||||||
11
0-4111
||||||||||
11|||||||||
0.5
|||||
1/ 111
III
- 1 | | | \ / |||||||
-11|| \/ ||
VIII
III
11-11
16
t

(b) Based on an inspection of the direction field, describe how solutions behave for large t.
O All solutions seem to approach a line in the region where the negative and positive slopes meet each other.
O All solutions seem to eventually have positive slopes, and hence increase without bound.
O All solutions seem to eventually have negative slopes, and hence decrease without bound.
O The solutions appear to be oscillatory.
O If y(0) > 0, solutions appear to eventually have positive slopes, and hence increase without bound. If y(0) ≤ 0, solutions appear to have negative slopes and decrease without bound.
(c) Find the general solution of the given differential equation.
y(t) =
Use it to determine how solutions behave as t → co.
7
O All solutions converge to the function y = cos 2t.
2
O All solutions will increase exponentially.
O All solutions converge to the function y =
2
7
O All solutions converge to the function y =
2
O All solutions will decrease exponentially.
sin 2t.

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