Solve the initial value problem x'(t) = Ax(t) for t≥0, with x(0)=(5,4). Classify the nature of the origin as an attractor, repeller,or saddle point of the dynamical system described by x'=Ax. Find the directions of greatest attraction and/or repulsion.A=x(t)=11 -13Solve the initial value problem.4DSolve the initial value problemx00=Classify the nature of the origin as an attractor, repeller, or saddle point. Choose the correct answer belowO RepellerO Saddle PointO AttractorChoose the correct graph below that represents the direction(s) of greatest attraction and/or repulsion: Choose the correct graph below that represents the direction(s) of greatest atraction and/or repulsionOAOB.Oc

Answered: Image /qna-images/answer/9ff33ad3-177e-4c0a-bf6c-1a730ced9f4c.jpg

Answered: Solve the initial value problem…

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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If y, and y, are linearly independent solutions of ty" + 3y' + tey = 0 and if W(y,, y2)(1) = 6, find W(y1, y2)(2). Round your answer to two decimal places. W(y1, y2)(2) = i AttomptcıO of E ucod
s). Consider the system 2 ²¹1) (3). - (a) Plot the solution curve passing through the point (*)-(6). = (b) Express the curve in the form y = F(x, y). [Hint: an integrating factor is (x² + y²)p.] (3)' = ( -2
Q2. (a) If b² – 4ac = 0 and y1 = emr solves ay" + by + cy = 0, use reduction of order to get a linearly independent solution. You may assume linear independence (6
r(t) = (-et+3, +³ - 8) 10. A particle moves through 3-space in such a way that its velocity is v(t) = ti+t²j+t³k. If the particle's initial position at time t = 0 is (1,2,3), what is the particle's position when t = 1? (Hint: set up an initial value problem.)
(14-17) find the solution of following derivative by Rung-Kutta method f(x,y)=-2x³ + 12x² - 20x + 8.5 let initial condition at x-0 is y-1 and h=0.5 F 14) y(0.5) - 15) M1 = 16) M2 .. 17) M4=
Which of the following functions V(y1, y2) is a Lyapunov function for the dynamical system with equilibrium point at (0, 0) y₁ = -2y₁y2e (12)² – 6y1, y2 = -2y/y₂e (19₂2) — 2y2 Select one: O a. V(y₁, y2) = y₁ + (y2 − 1)² O b. O c. O d. V(y₁, y₂) = y²e(9192)² + y² − 3y₁ - V(y₁, y2) = e(3132)² V(y₁, y2) = e(1¹³₂)² + y² − 1 + 3y²
Which of the following functions V(y1, Y2) is a Lyapunov function for the dynamical system y1 = -2y1yževí v½ – Y1, ý2 = -2yjyzeyi vi Select one: O a. V(y1, Y2) = e1 ½ + O b. V(y1, Y2) = evi v? + }yi – 1 O c. V(y1, Y2) =-y1y2eiv½ – y? O d. V(y1, 42) = -evi v? +
Which of the following functions V(y₁, y2) is a Lyapunov function for the dynamical system with equilibrium point at (0, 0) y₁ = −2y₁y¾e(Y192)² – 6y₁, y₂ = -2y²y₂e(³₁V2)² – 2y₂ Select one: a. ○ b. V(y₁, y2) = e(Y1Y2)² V(y₁, y2) = e(¹¹³₂)² + y² −1+3y² ○ c. V(y₁, ₂) = y²e(₁8₂)² + y² − 3y₁ O ○ d. V(y₁, y₂) = y† + (y₂ − 1)²
5. Determine whether the functions y, and y2 are linearly dependent on the interval (0,1). Yı = 1+ e', Y2 = 1 – 2e2t
Find the equilibrium solutions and determine which are stable and which are unstable. y' = 9y – 3y? O y=-3 (stable); y = 0 (unstable) y = 0 (unstable); y=3 (stable) y = 0 (stable); y = 3 (unstable) y=-3 (unstable); y = 0 (stable)
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= 1. A particle 1 of mass m₁ is attached to one end of a spring with constant k at position x = = 0. It is initially at rest, with velocity u₁ 0. The other end of the spring is attached to a fixed wall at position x = -l where l is the length of the spring at rest. The whole setup is horizontal along the x-axis. Another particle, 2, of mass m2 travels towards particle 1 with constant velocity u2 (note that u2 < 0 in this setup). The two particles undergo a collision with coefficient of restitution 0 ≤ e ≤ 1. (a) Using the one-dimensional equations of collision (eq (6.44) in the lecture notes) show that immediately after collision the velocity of particle 1 is given by V₁ = (1 + e) m2u2 m1 + m2 Find the velocity v2 of particle 2 immediately after collision. What happens to particle 2 if m₁ = m2 and the collision is elastic (e = 1)? (b) Immediately after collision, particle 1 has velocity v₁ and is at position x = 0, which we take as initial conditions for its subsequent motion under the…

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