 9.2.1: In each of 1 through 3, use an appropriate graphing device to draw ...
 9.2.2: In each of 1 through 3, use an appropriate graphing device to draw ...
 9.2.3: In each of 1 through 3, use an appropriate graphing device to draw ...
 9.2.4: For each of the systems in 4 through 13: a. Find all the critical p...
 9.2.5: For each of the systems in 4 through 13: a. Find all the critical p...
 9.2.6: For each of the systems in 4 through 13: a. Find all the critical p...
 9.2.7: For each of the systems in 4 through 13: a. Find all the critical p...
 9.2.8: For each of the systems in 4 through 13: a. Find all the critical p...
 9.2.9: For each of the systems in 4 through 13: a. Find all the critical p...
 9.2.10: For each of the systems in 4 through 13: a. Find all the critical p...
 9.2.11: For each of the systems in 4 through 13: a. Find all the critical p...
 9.2.12: For each of the systems in 4 through 13: a. Find all the critical p...
 9.2.13: For each of the systems in 4 through 13: a. Find all the critical p...
 9.2.14: In each of 14 through 20: a. Find an equation of the form H( x, y) ...
 9.2.15: In each of 14 through 20: a. Find an equation of the form H( x, y) ...
 9.2.16: In each of 14 through 20: a. Find an equation of the form H( x, y) ...
 9.2.17: In each of 14 through 20: a. Find an equation of the form H( x, y) ...
 9.2.18: In each of 14 through 20: a. Find an equation of the form H( x, y) ...
 9.2.19: In each of 14 through 20: a. Find an equation of the form H( x, y) ...
 9.2.20: In each of 14 through 20: a. Find an equation of the form H( x, y) ...
 9.2.21: Given that x = (t), y = (t) is a solution of the autonomous system ...
 9.2.22: Prove that for the system dx dt = F( x, y), dy dt = G( x, y) there ...
 9.2.23: Prove that if a trajectory starts at a noncritical point of the sys...
 9.2.24: 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 and Boundary Value Problems  11th Edition
ISBN: 9781119256007
Solutions for Chapter 9.2: Autonomous Systems and Stability
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007. Since 24 problems in chapter 9.2: Autonomous Systems and Stability have been answered, more than 12406 students have viewed full stepbystep solutions from this chapter. Chapter 9.2: Autonomous Systems and Stability includes 24 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Iterative method.
A sequence of steps intended to approach the desired solution.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·