parts (abcd) of 1 AEXERCISES1. For each of the following nonlinear systems: (a) Find all the critical points.(b) Use the Jacobian matrix to classify the critical points (saddle, node,etc., stable, asymptotically stable, etc.) (c) In the phase plane sketch thenullclines and indicate the direction field in regions bounded by the null-clines. (d) Insofar as possible, sketch a rough phase diagram indicatingseveral key orbits.a) I' = x - y, y' = 2x(-1+y).

The given system of differential equations are x'=x-yy'=2x-1+y

Answered: parts (abcd) of 1 A EXERCISES 1. For…

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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and x® (t) = (). t (2) Consider the vectors x(t) = ( 10) (a) Compute the Wronskian of x(1) and x(2). X W (b) In what intervals are x1) and x2) linearly independent? D (c) What conclusion can be drawn about coefficients in the system of homogeneous differential equations satisfied by x1) and x2)? Choose one ▼ of the coefficients of the ODE in standard form must be discontinuous at to (d) Find the system of equations x' = P(t)x. P(t) =
The function of 2 variables: f (x,y) = (5/2)x2 + 2xy - 12x + (3/2)y2 - 7y + (31/2) Has a stationary point at (2,1). At this stationary point, the Hessian matrix for f is: H = [5 2] [2 3] Which one of these options best describes the stationary point? A. The stationary point is a saddle point B. The stationary point is a local minimum C. The stationary point is a local maximum D. The nature of the stationary point cannot be determined by the Hessian matrix
Find the general solution in terms of real functions. (b) From the roots of the characteristic equation, determine whether each critical point of the corresponding dynamical system is asymptotically stable, stable, or unstable, and classify it as to type. (c) Use the general solution obtained in part (a) to find a two-parameter family of trajectories x=x1i+x2j=yi+y′j of the corresponding dynamical system. Then sketch by hand, or use a computer, to draw a phase portrait, including any straight-line orbits, from this family of trajectories.
a) Which of the following systems of ODEs is autonomous and dynamical? III, II where 1: y₁= e = sin(y2), y2 = ₁ + y2 II: dy dy₁ dx dy₂ = 5y₁, y = dx Y₂' = V1 Y2 - y2 1 III: Y1 = Зуг, = -12 b) Find out which of the following options are equilibria of the dynamical system, IV where I:(y₁, y) = (5,0); II:(y₁, y₂) = (3,0); III:(y₁, y) = (0, 1); IV:(y₁, y) = (3,2). y₁ = ₁₂ - 1, y2 = (y₁ - 3)y2 =
7. (a) Find the equation of the straight line y = ao + a₁t which best fits the following points: 12 13 4 t y 1 3 4 2 2 3 (b) Find the projection matrix onto C(A), the column space of A, where A = 1 1 1 2 1
7. Consider the following nonlinear first order system of equations defined for x E R, y E R: x' -3x + cos Y (a) Find all the equilibrium points of the system. (b) Compute the Jacobian of the sytem. J(x, y) : (c) Use the Jacobian and the Trace-Determinant plane to classify each equilibrium point (spiral source, nodal sink, etc.)
I don't understand what I am missing with the critical points.
The function of two variables ƒ(x, y) = 3x² – 2xy − 10x + y² + 2y +9 has a stationary point at (2, 1). At this stationary point, the Hessian matrix for fis H=[6₂ -2 Select the option that describes this stationary point. Select one: 2]. -2 The stationary point is a saddle point. The nature of the stationary point cannot be determined from the Hessian matrix. The stationary point is a local maximum. The stationary point is a local minimum.

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