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6. Consider the system x' = 2r y=-y. I (a) Sketch the vector field. (b) Show that for every solution (r(t), y(t)) there exists a constant C such that the solution lies on the curve ay²= C. Find C in terms of ro= r(0) and yo-y(0). (e) Sketch several typical solutions and describe their behavior as t-oc and as f→-00.

6. Consider the system x' = 2r y=-y. I (a) Sketch the vector field. (b) Show that for every solution (r(t), y(t)) there exists a constant C such that the solution lies on the curve ay²= C. Find C in terms of ro= r(0) and yo-y(0). (e) Sketch several typical solutions and describe their behavior as t-oc and as f→-00.

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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6. Consider the system a' = 2r y=-y. I (a) Sketch the vector field. (b) Show that for every solution (z(t), y(t)) there exists a constant C such that the solution lies on the curve ay² C. Find C in terms of zo z(0) and yo-y(0). = (e) Sketch several typical solutions and describe their behavior as too and as t→-00.
Transcribed Image Text:6. Consider the system a' = 2r y=-y. I (a) Sketch the vector field. (b) Show that for every solution (z(t), y(t)) there exists a constant C such that the solution lies on the curve ay² C. Find C in terms of zo z(0) and yo-y(0). = (e) Sketch several typical solutions and describe their behavior as too and as t→-00.
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