Consider a planar gradient vector field with V(, 2- 1)2 2. For this problem, (a) Obtain the ODE (...
1. (This is problem 5 from the second assignment sheet, reprinted here.) Consider the nonlinear system a. Sketch the ulllines and indicate in your sketch the direction of the vector field in each of the regions b. Linearize the system around the equilibrium point, and use your result to classify the type of the c. Use the information from parts a and b to sketch the phase portrait of the system. 2. Sketch the phase portraits for the following systems...
Problem 2.Given the following dynamic system Given the Lyapunov (energy) function: V = 1. What is the definiteness (positive definite PD, negative definite ND, PSD, NSD) of? 2. What is the definiteness of V - dl 3. Based on Lyapunov Stability theorem, is the system stable? 4. Using the eigenvalues technique, is the system stable? dt Problem 2.Given the following dynamic system Given the Lyapunov (energy) function: V = 1. What is the definiteness (positive definite PD, negative definite ND,...
(1 point) Math 215 Homework homework9, Problem 2 Find the gradient vector field of the function f(x, y) = -75x2 + y2. F(x,y) =
Previous Problem Problem List Next Problem (1 point) Show that the vector field F(x, y, z) show what you intended? (-3y cos(5x), 5x sin(-3y),0) is not a gradient vector field by computing its curl. How does this curl(F) = V × F-《
Problem 2 Obtain the second-order ODE describing the capacitor voltage v(t) in the series RLC circuit shown below. Hint: Confer with Problem 3.14 in the textbook and use i()for the loop current. 1S2 1 H v(t) (t 2F)
vector x' = [ the first row is 2 and 8, the second row is -1 and -2] vector x (i) Compute the eigenvalues and eigenvectors of the system. (ii) Use the eigenvalues to classify the equilibrium type of the origin. (iii) Use the eigenvectors as guides to plot a phase portrait of the system. (iv) Present a general solution to the system of ODE. (v) Find the particular solution to this system of ODE if vector x(0) = [...
1 1 Consider the function f(x.y,z) 2x y 2 the point P(3,0,1), and the unit vector u 0 Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P c. Find the rate of change of the function in the direction of maximum increase at P d. Find the directional derivative at P in the direction of the given vector. a. 1 1 Consider the function...
the excercise concerns the function (x^2 + y^2)* e^(1-x^2 - y^2) please do all parts MA330 Homework #4 1. This exercise concerns the function its gradient vector field F-vo See the plots of each below. a) Compute the partial derivatives os and ty to find the gradient field vo. (b) In MA231, learned 1, you learned that mixed second-order partial derivatives of reasonable functions Verity that here by computing day and dys and checking that they are the same. should...
PROBLEM #2: Consider the metric: and vector: a) Compute the length-squared of the vector v. b) Using Mathematica, make a ContourPlot of the length of the vector in the x,y space. Comment about the result. Note, for this one, just focus on the first quadrant; I'll explain in class. c) Can you think of a physical example where this is the appropriate metric to use to measure distance??? There are more than one answer, but the more nat ural the...
Do problem 5.6 a. Obtain a complete SSR with input u and output h. Derive the system transfer function Go) Zs/u c. Derive the transfer function Y(s)/U(s) where the output is y Obtain a complete SSR for the given system, with input u = v and output 5.3 0.25t +2c-0.6w = 0 5.4 Given the nonlinear first-order system Derive the linear model by performing the linearization about the static equilibrium state a that res when the nominal input is "....