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. Consider the following non-linear system of differential equations x'=xy- 2, y = x - 2y. Note that (2, 1) is a stationary point of the vector field associated with the above equation. Problem 1 (a) A stationary point (x, y) of a vector field is called isolated if there exists a neighborhood U around (x, y) such that (x, y) be the only stationary point in U. Show that (1, 2) is an isolated stationary point of the vector field associated with the above system. (b) Find a suitable change of coordinates that maps (2, 1) to (0, 0), and write the linearization of the new system. (c) Sketch the local phase portrait of the new system near the point (0,0). (d) A critical point (x, y) is called asymptotically stable whenever there exists >0 such that the following holds if √√√(x(0) − x)² + (y(0) — y.)² < 6, Use the phase portrait that you sketched in part (c) to see whether the point (0, 0) is asymptotically stable or not. then lim (x(t), y(t)) = (x*, Y*). t-x

. Consider the following non-linear system of differential equations x'=xy- 2, y = x - 2y. Note that (2, 1) is a stationary point of the vector field associated with the above equation. Problem 1 (a) A stationary point (x, y) of a vector field is called isolated if there exists a neighborhood U around (x, y) such that (x, y) be the only stationary point in U. Show that (1, 2) is an isolated stationary point of the vector field associated with the above system. (b) Find a suitable change of coordinates that maps (2, 1) to (0, 0), and write the linearization of the new system. (c) Sketch the local phase portrait of the new system near the point (0,0). (d) A critical point (x, y) is called asymptotically stable whenever there exists >0 such that the following holds if √√√(x(0) − x)² + (y(0) — y.)² < 6, Use the phase portrait that you sketched in part (c) to see whether the point (0, 0) is asymptotically stable or not. then lim (x(t), y(t)) = (x*, Y*). t-x

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 non-linear system of differential equations x' = xy - 2, y' = = X 2y. Note that (2, 1) is a stationary point of the vector field associated with the above equation. Problem 1 ● (a) A stationary point (x, y) of a vector field is called isolated if there exists a neighborhood U around (x, y) such that (x, y) be the only stationary point in U. Show that (1,2) is an isolated stationary point of the vector field associated with the above system. (b) Find a suitable change of coordinates that maps (2, 1) to (0, 0), and write the linearization of the new system. (c) Sketch the local phase portrait of the new system near the point (0,0). (d) A critical point (x, y) is called asymptotically stable whenever there exists > 0 such that the following holds if_√(x(0) − xx)² + (y(0) − y+)² < 8, Use the phase portrait that you sketched in part (c) to see whether the point (0, 0) is asymptotically stable or not. then _lim (x(t), y(t)) = (x*, Y*). t→∞
Transcribed Image Text:Consider the following non-linear system of differential equations x' = xy - 2, y' = = X 2y. Note that (2, 1) is a stationary point of the vector field associated with the above equation. Problem 1 ● (a) A stationary point (x, y) of a vector field is called isolated if there exists a neighborhood U around (x, y) such that (x, y) be the only stationary point in U. Show that (1,2) is an isolated stationary point of the vector field associated with the above system. (b) Find a suitable change of coordinates that maps (2, 1) to (0, 0), and write the linearization of the new system. (c) Sketch the local phase portrait of the new system near the point (0,0). (d) A critical point (x, y) is called asymptotically stable whenever there exists > 0 such that the following holds if_√(x(0) − xx)² + (y(0) − y+)² < 8, Use the phase portrait that you sketched in part (c) to see whether the point (0, 0) is asymptotically stable or not. then _lim (x(t), y(t)) = (x*, Y*). t→∞
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