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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
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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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
Transcribed Image Text: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.
Transcribed Image Text:(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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