Consider6x+ 2y(1+z²)²2(x+y)(1+z²)²+ y? show that the origin is asymptotically stable andUsing the function V(x, y)determine a large subset of its basin of attraction.=1+x?

What is Lyapunov Stability Theorem: Lyapunov stability theorem says that if a function is positive…

Answered: Consider 6x (1+x²)² 2(x+y) (1+x²)² + 2y…

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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Find a simple closed curve C with counterclockwise orientation that maximizes the value of dx -**) + O Cisa circle of radius 2 with the center at (2, 2). O Cis the triangle with vertices (0, 0), (2, 0), and (0, 2). O Cis the square with vertices (0, 0), (2, 0), (2, 2), and (0, 2). O The integral doesn't depend on the form of C. O Cisa circle of radius 2 with the center at the origin.
(b) Let C₁, C₂2 and C3 be the curves in R³ with parametrisations C₁: r₁(s) (s − 2)i + (2s² − 1)j + k (SER) C₂ r₂(t) 2t² i + (3t+7)j + (t+1) k (t = R) C3 r3(u) (u² + 4u+3)i + (u² + 6)j + (u + 2)k (u ¤R) = = = respectively. (i) Show that C₁, C₂ and C3 intersect at the point (0, 7, 1). answer. (ii) Find the tangent vectors to the curves C₁, C₂ and C3 at the point (0, 7, 1). Do these tangent vectors lie in a common plane in R³? Justify your (iii) Is it possible to find a surface z = h(x, y) for which the curves C₁, C₂ and C3 all lie on this surface? Justify your answer.
Consider the functionf(x, y, z) =z² i+y cos(x) j +y sin (x) k........
(b) Let C₁, C₂ and C3 be the curves in R³ with parametrisations C₁: r₁(s) (s − 2)i + (2s² − 1)j + k - C₂ r₂(t) C3 r3(u) respectively (i) = = (SER) 2t² i + (3t+7)j + (t+1) k (t = R) (u² +4u + 4) i + (u² + 6)j + (u + 2) k (u ¤R) giving that the point of intersection exist and just asking you to verify that the curves intersect at a point (ii) Find the tangent vectors to the curves C₁, C2 and C3 at the unique point of intersection you find in part (i). Do these tangent vectors lie in a common plane in R³? Justify your answer. (iii) Is it possible to find a surface z = h(x, y) for which the curves C₁, C₂ and C3 all lie on this surface? Justify your answer.
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Consider 6x (1+x²)² 2(x+y) (1+x²)² + 2y ý = Using the function V(x, y) = 12 + y² show that the origin is asymptotically stable and determine a large subset of its basin of attraction. 1+z²