Consider the system described by x˙(t) = u(t) − sin (x(t)) y(t) = u(t) + cos (x(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.

Find a general solution to each linear ODE, and then find the specific solution with the given initial condition by using the integrating factor (a) u'(t) = u(t)+3, u(0) = 3 (b) u'(t) = 2u(t)+4, u(0) = 0 (c) u'(t) = −3u(t) +3, u(0) = 5 (d) u'(t) = −3u(t) +9t, u(0) = 5 (e) u' (t) = u(t)+2sin(t), u(0) = 1 (f) u'(t) = −4u(t) + e¹, u(0) = 2 (g) u'(t)= tu(t)+t, u(0) = 2 (h) u' (t) = u(t)/t+2, u(1) = 3 (i) u' (t) = sin(t)u(t) + sin(t), u(0) = 4 (j) u' (t) = au(t) +b, u(0) = uo, where a, b, and uo are constants

1. Given z = e' tan(x) + (2x + 3y)5 x=71² y = 5t +9 (a) Draw a tree diagram showing which variables depend on which. The variable z should be at the top. (b) Calculate the partial derivatives of z. Əz dx (c) Calculate the derivatives of x(t) and y(t). dx dt dy dt dz dt = (d) Use the multivariable chain rule to calculate || dz dt || z(t) = əz ду (e) Now plug in for x and y in terms of t to find z as a function of t, then differentiate that function. =

Let x(t) and y(t) satisfy the system of differential equations dx За(1 — х — 4у) dt dy = .2y((1 – .5x – y) dt (a) If x(4) = 5 and y(4) = 6 find x'(4) and y'(4). (b) Is y(t) increasing or decreasing when t = 4, explain how you know. (c) Estimate x(4.05) and y(4.05).