Find and classify the equilibrium points of the system dx xy + x – 2y = dt dy (2² – 1)(y – 3) | dt
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- find the first order linear y(t)=4et-1-t-1Consider the ecuation ky" – ky' – 210y = 0, with constant real k + 0. If one - %3D solution of the ecuation is: Y1(x) = e then another solution linearly %3D dependent of the equation with y1 is: O a) y2(z) = el7z O b) y2(x) = ekz Oc) y2(x) = e¬1llz %3D O d) y2(x) = e-5z %3DWhich of the following functions V(y1, Y2) is a Lyapunov function for the dynamical system yi = -(y1 – y2)e(un -)* Ý2 = (y1 – y2)e(% 4)* – 2y? %3D Select one: a. V(y1, Y2) = e(y-4)² O b. V(y1, Y2) = y5 O c. V(y1, Y2) = -e-92)* + y – 1 + y – 1 d. V(y1, Y2) = elon 2)² * – y Y2
- 6. Consider the dynamical system dx - = x (x² − 4x) - dt where X a parameter. Determine the fixed points and their nature (i.e. stable or unstable) and draw the bifurcation diagram.Which of the following functions V(y1, y2) is a Lyapunov function for the dynamical system y₁ = (4- y₁)e³₁³², y2 = y₁ - y₁y2 Select one: O a. V(y₁, y₂) = (y₁ - 4)4+ (y2 - 1)² O b. V(y₁, 2) = y + y²/2 O c. V(y₁, 2) = −y + (y₂ - 1)³ 1 O d. V(y₁, 2) = −₁+e%2²Get all the balance points -x1 + 2xi + x2 -x1 - x2 %3D F = {F1,72} Obtain all the equilibrium points of the dynamical system. a. ī = {[0,0]} b.ī = {[1, }],, +1]} c. there are infinite points of balance d. = {[0,0], [1, –1], [–1, +1]} {[0, 0], [1, –k], [–k, +1]}Vk € Z+ e. i =
- Q2:B. Consider a continuous-time system with the input-output relation: y(t) = T{x(t)} = x(t)8(t – KT,) Determine whether this system is (a) linear, (b) time-invariant.Which of the following functions V(y1, y2) is a Lyapunov function for the dynamical system y₁ = (4-₁)ex, y2 = y²-y²y₂ Select one: a. V(v₁.32) = -4+(2-1)³ b. V(v₁.y2) = -²³ +²² c. V(v₁.92) = ₁ + y/² C. Od. V(y₁, y2) = (y₁ - 4)4+ (y2 - 1)²Determine two linearly independent solutions to x2y′′+xy′−(4+x)y =0, x > 0.
- Find the linearization at x = a. f(x) = ) = — - ₁a = 1 x2, = 7Which of the following functions V(y₁, y2) is a Lyapunov function for the dynamical system y₁ = (4 − y₁)e³¹³², y2 = y² - y²y₂ Select one: a. V(y₁, y₂) = (y₁ −4)4 + (y₂ − 1)² b. V(y₁, y₂) = y₁ + y² c. V(y₁, y₂) = −y} +e³ d. V(y₁, y₂) = −y + (y₂ − 1)³