 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 28333 students have viewed full stepbystep solutions from this chapter. Elementary Differential Equations was written by and is associated to the ISBN: 9780470458327.

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

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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).

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

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.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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

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