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dx 8y – x² – 12 dt - | dy - 2(x – 2)(y – 6) dt (a) Does there exist a yo > 0 such that if x(0) > 0 and y(0) > yo, then y(t) > yo for all t > 0? If so, give such a yo and explain briefly. If not, justify your answer.

dx 8y – x² – 12 dt - | dy - 2(x – 2)(y – 6) dt (a) Does there exist a yo > 0 such that if x(0) > 0 and y(0) > yo, then y(t) > yo for all t > 0? If so, give such a yo and explain briefly. If not, justify your answer.

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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dx 8y — г2 — 12 dt - dy = 2(x – 2)(y – 6) dt %3D - | (a) Does there exist a yo > 0 such that if x(0) > 0 and y(0) > Yo, then y(t) > yo for all t > 0? If so, give such a yo and explain briefly. If not, justify your answer. (b) Is it true that, if y(0) <((x(0))² + 12), then we will have y(t) <((x(t))² + 12) for all t > 0? (c) Solve the system in the special case where y(0) = 6 and x(0) # 6, that is, find y(t) and x(t) for all t > 0. (This requires techniques from previous units!)
Transcribed Image Text:dx 8y — г2 — 12 dt - dy = 2(x – 2)(y – 6) dt %3D - | (a) Does there exist a yo > 0 such that if x(0) > 0 and y(0) > Yo, then y(t) > yo for all t > 0? If so, give such a yo and explain briefly. If not, justify your answer. (b) Is it true that, if y(0) <((x(0))² + 12), then we will have y(t) <((x(t))² + 12) for all t > 0? (c) Solve the system in the special case where y(0) = 6 and x(0) # 6, that is, find y(t) and x(t) for all t > 0. (This requires techniques from previous units!)
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