Which of the following functions V(y1, y2) is a Lyapunov function for the dynamical system with equilibrium point at (0,0)y₁ = −2y₁y²e(¹₁³₂)² – 6y₁, y2 = -2y²y₂e³1³2)² – 2y2Select one:○a. V(y₁, y₂) = e(4₁3₂)².O b.V(y₁, 2) = e(1₁1₂) ²O c.V(y₁, y2) = y₁ + (y₂ − 1)²O d.V(y₁, y2) = ye(³1½)² + y² − 3y₁+ y −1+3y²

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Answered: Which of the following functions V(31,…

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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Enter your ans wers as decimal numbers. 2y" +8y' : The general solution of -6y y(t): C1e"it + C2e"2t, where can be written as |r1| < |r2|. The values of T1 and r2 are, respectively, and y(0) = 2 and y'(0) = –1, then If C1 = and C2 and y(5) = (Round this answer to two decimal places.)
Which of the following functions V(y₁, y2) is a Lyapunov function for the dynamical system with equilibrium point at (0, 0) y₁ = −2y₁y²e(1³2)² – 6y₁, y₂ = −2y²y₂e(1³2)² – 2y2 V2 Select one: O a. V(y₁, y₂) = e(1₁³2)² ○ b. c. O d. V(y₁, Y2) = e(1¹³2)² + y² −1+3y² V(y₁, y2) = y²e (₁³2)² + y² − 3y₁ - V(y₁, y₂) = y₁ + (y₂ − 1)²
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Consider the system described by i(t) = u(t) – sin (r(t)) y(t) = u(t) + cos (r(t)) a) Find all equilibrium points of the system. b) For each equilibrium point, determine whether or not the equilibrium point is (i) stable in the sense of Lyapunov; (ii) asymptotically stable; (iii) globally asymptotically stable. Explain your answers. c) Determine whether or not the system is bounded-input bounded-output stable. Explain your answer.
1. Consider the following partial differential equation: du (T,y) -3 (r, y) + 2u(r, y) = 0....(E). Let a and b be two real numbers. For which values of a and b, the function u(x, y) = f(y)ear + g(y)eb" is a solution of (E), where f(y), g(y) are two arbitrary functions. a) a = 2 and b = 3 b) a = 1 and b= 2 c) a = -2 and b = -1 d) None of the above. 2. Using the characteristic method, the solution of the following partial differential equation: |ru + (y+1)uy = u |u(x,0) = f(x), %3D where f is a given function. a) u(r, y) = (y + 1)f ( b) u(x, y) = (y² + 1)f( ). c) u(r, y) = f() d) None of the above. 3. The partial differential equation uz.Urr+ Urr.Uyy – u -r = 0 is a) Second order quasi-linear non-homogenous equation. b) Second order quasi-linear homogenous equation. c) Second order homogenous linear equation. d) None of the above.
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3. Find a fourth order homogeneous linear ODE with solution x(t) = 3t2 – 1+6e¬4t. -

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