Solve the given nonlinear plane autonomous system by changing to polar coordinates. x' = y - y' = -x - X 2 V x² + (r(t), 0(t)) = X(0) = = (1, 0) (r(t), 0(t)) = y V x² + (64 - x² - y²) (64 - x² - y²), (solution of initial value problem) Describe the geometric behavior of the solution that satisfies the given initial condition. The solution approaches the origin on the ray 0 = 0 as t increases. The solution spirals toward the circle r = 8 as t increases. The solution traces the circle r = 8 in the clockwise direction as t increases. The solution spirals away from the origin with increasing magnitude as t increases. The solution spirals toward the origin as t increases. X(0) = = (8,0) (solution of initial value problem) Describe the geometric behavior of the solution that satisfies the given initial condition. The solution approaches the origin on the ray 0 = 0 as t increases. The solution spirals toward the circle r = 8 as t increases. The solution traces the circle r = 8 in the clockwise direction as t increases. The solution spirals away from the origin with increasing magnitude as t increases. The solution spirals toward the origin as t increases.
Solve the given nonlinear plane autonomous system by changing to polar coordinates. x' = y - y' = -x - X 2 V x² + (r(t), 0(t)) = X(0) = = (1, 0) (r(t), 0(t)) = y V x² + (64 - x² - y²) (64 - x² - y²), (solution of initial value problem) Describe the geometric behavior of the solution that satisfies the given initial condition. The solution approaches the origin on the ray 0 = 0 as t increases. The solution spirals toward the circle r = 8 as t increases. The solution traces the circle r = 8 in the clockwise direction as t increases. The solution spirals away from the origin with increasing magnitude as t increases. The solution spirals toward the origin as t increases. X(0) = = (8,0) (solution of initial value problem) Describe the geometric behavior of the solution that satisfies the given initial condition. The solution approaches the origin on the ray 0 = 0 as t increases. The solution spirals toward the circle r = 8 as t increases. The solution traces the circle r = 8 in the clockwise direction as t increases. The solution spirals away from the origin with increasing magnitude as t increases. The solution spirals toward the origin as t increases.
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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Transcribed Image Text:Solve the given nonlinear plane autonomous system by changing to polar coordinates.
x' = y -
y' = -x -
X
2
V x² +
(r(t), 0(t)) =
X(0) = = (1, 0)
(r(t), 0(t)) =
y
V x² +
(64 - x² - y²)
(64 - x² - y²),
(solution of initial value problem)
Describe the geometric behavior of the solution that satisfies the given initial condition.
The solution approaches the origin on the ray 0 = 0 as t increases.
The solution spirals toward the circle r = 8 as t increases.
The solution traces the circle r = 8 in the clockwise direction as t increases.
The solution spirals away from the origin with increasing magnitude as t increases.
The solution spirals toward the origin as t increases.
X(0) = = (8,0)
(solution of initial value problem)
Describe the geometric behavior of the solution that satisfies the given initial condition.
The solution approaches the origin on the ray 0 = 0 as t increases.
The solution spirals toward the circle r = 8 as t increases.
The solution traces the circle r = 8 in the clockwise direction as t increases.
The solution spirals away from the origin with increasing magnitude as t increases.
The solution spirals toward the origin as t increases.
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