 9.2.1: In each of 1 through 4, sketch the trajectory corresponding to the ...
 9.2.2: In each of 1 through 4, sketch the trajectory corresponding to the ...
 9.2.3: In each of 1 through 4, sketch the trajectory corresponding to the ...
 9.2.4: In each of 1 through 4, sketch the trajectory corresponding to the ...
 9.2.5: For each of the systems in 5 through 16:(a) Find all the critical p...
 9.2.6: For each of the systems in 5 through 16:(a) Find all the critical p...
 9.2.7: For each of the systems in 5 through 16:(a) Find all the critical p...
 9.2.8: For each of the systems in 5 through 16:(a) Find all the critical p...
 9.2.9: For each of the systems in 5 through 16:(a) Find all the critical p...
 9.2.10: For each of the systems in 5 through 16:(a) Find all the critical p...
 9.2.11: For each of the systems in 5 through 16:(a) Find all the critical p...
 9.2.12: For each of the systems in 5 through 16:(a) Find all the critical p...
 9.2.13: For each of the systems in 5 through 16:(a) Find all the critical p...
 9.2.14: For each of the systems in 5 through 16:(a) Find all the critical p...
 9.2.15: For each of the systems in 5 through 16:(a) Find all the critical p...
 9.2.16: For each of the systems in 5 through 16:(a) Find all the critical p...
 9.2.17: In each of 17 through 24:(a) Find an equation of the form H(x, y) =...
 9.2.18: In each of 17 through 24:(a) Find an equation of the form H(x, y) =...
 9.2.19: In each of 17 through 24:(a) Find an equation of the form H(x, y) =...
 9.2.20: In each of 17 through 24:(a) Find an equation of the form H(x, y) =...
 9.2.21: In each of 17 through 24:(a) Find an equation of the form H(x, y) =...
 9.2.22: In each of 17 through 24:(a) Find an equation of the form H(x, y) =...
 9.2.23: In each of 17 through 24:(a) Find an equation of the form H(x, y) =...
 9.2.24: In each of 17 through 24:(a) Find an equation of the form H(x, y) =...
 9.2.25: Given that x = (t), y = (t) is a solution of the autonomous systemd...
 9.2.26: Prove that for the systemdx/dt = F(x, y), dy/dt = G(x, y)there is a...
 9.2.27: Prove that if a trajectory starts at a noncritical point of the sys...
 9.2.28: Assuming that the trajectory corresponding to a solution x = (t), y...
Solutions for Chapter 9.2: Autonomous Systems and Stability
Full solutions for Elementary Differential Equations  10th Edition
ISBN: 9780470458327
Solutions for Chapter 9.2: Autonomous Systems and Stability
Get Full SolutionsChapter 9.2: Autonomous Systems and Stability includes 28 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10. This expansive textbook survival guide covers the following chapters and their solutions. Since 28 problems in chapter 9.2: Autonomous Systems and Stability have been answered, more than 9377 students have viewed full stepbystep solutions from this chapter. Elementary Differential Equations was written by and is associated to the ISBN: 9780470458327.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.