[a z For the system x+1) investigate the type of the equilibrium points (0,0) and (1.-1. [a z For the system x+1) investigate the type of the equilibrium points (0,0) and (1.-1.
7. [MT, p. 210] Investigate whether or not the system u(x, y, z) = x + xyz V(x, y, z) y + xy W(x, y, z) = 2 + 2x + 322 = can be solved for x, y, z in terms of u, v, w near (x, y, z) = (0,0,0).
5. Determine the possible equilibrium states of the system and investigate their stability by the first method of Lyapunov, if it is described by a system of AN = P(x, y); 7 = Q(x, y) equations p dt P(x, y) = 2x²y +1 Q(x, y) = x + y
Define f: R2R by 224V2 y) (0,0) 0 if (x, y)-(0,0) if (z, f(z, y) (a) Prove that Dif(z, y) and D2f (x, y) exist for each (x, y) E R2. (b) Prove that f is not continuous at (0,0).
Consider the system: x' = y(1 + 2x) y' = x + x2 - y2 a. Find all the equilibrium points, and linearize the system about each equilibrium point to find the type of the equilibrium point. b. Show that the system is a gradient system, and conclude that it has no periodic solutions. c. Sketch the phase portrait. Explain how you determined what the phase portrait looks like.
1. Classify the stability type of the equilibrium point (0,0) of the following linear planar homogeneous systems (same notation as in the previous homework). -(G3) 1 -3 2 -1 A-(25 4 -5
2. (2 pts) Determine the type of the critical point (0,0) for the system x' =-7x+ 5y, y' =-6x 4y. Sketch a phase portrait based on the eigenvectors, and the direction that the sign of the eigenvalue indicates. 2. (2 pts) Determine the type of the critical point (0,0) for the system x' =-7x+ 5y, y' =-6x 4y. Sketch a phase portrait based on the eigenvectors, and the direction that the sign of the eigenvalue indicates.
3. (10 points) The life of an electronic equipment is a r.v. X whose p.d.f is f(z;θ)-Be 4xz > 0,0>0, and let be its expected lifetime. On the basis of the random sample Xi,..X from this distribution, derive the MP test for testing the hypothesis Ho:to against the alternative HA:-(>o) at level of significance o 3. (10 points) The life of an electronic equipment is a r.v. X whose p.d.f is f(z;θ)-Be 4xz > 0,0>0, and let be its expected...
2. Define f : RR by - y 1(x) = { "2+2 (ay) (0,0); (z,y) = (0,0). (i) Isf continuous at (0,0)? Justify your answer. (ii) Show that Daf(3,0) = x for all x and D.f(0,y) = -y for all y (iii) D2f(0,0) + D2,1f(0,0). (iv) Is f differentiable at (0,0)? Justify your answer.
Please do the parts in the given order tyā (x,y)メ(0,0) (x,y)= (0,0). if if 1 (d) Given the unit vector u-( find the directional derivative of f(x, y) at the 리지, ,- point (to,m) = (0,0), in the direction of the vector a. (e) Find the gradient of f(x, y) at the point (zo,o) (0,0) (c) Find the equation of the tangent plane to the graph of the function z -f(x, y) at the point (x,y,z) (1,0,0). tyā (x,y)メ(0,0) (x,y)=...
1. Consider the system 2(t)--3i(t) +z2(t) +3() (a) (i) Find the linearised system at the equilibrium point (0, 0). (ii) What type of equilibrium point is (0,0)? (State your reasons fully.) (ii) Sketch the phase portrait for the linearised system near (0,0). (b) Repeat part (a) for the equilibrium point at (1,0). (c) (i) Are there any other equilibria? (ii) Read the Grobman-Hartman theorem and confirm that it applies to the above equilibria. 1. Consider the system 2(t)--3i(t) +z2(t) +3()...