 4.6.1: Show that all solutions x(t), y(t) of which start in the first quad...
 4.6.2: Show that all solutions x(t), y(t) of  dx dL =y(eX I),  =x+eY dt...
 4.6.3: Show that all solutions x(t), y(t) of which start in the upper half...
 4.6.4: Show that all solutions x(t), y (t) of which start inside the unit ...
 4.6.5: Let x(t), y(t) be a solution of with x(to)#y(to). Show that x(t) ca...
 4.6.6: Can a figure 8 ever be an orbit of where f and g have continuous pa...
 4.6.7: Show that the curve y2 + x2 + x4/2 = 2c2 is closed. Hint: Show that...
 4.6.8: Prove that all solutions of the following secondorder equations ar...
 4.6.9: Prove that all solutions of the following secondorder equations ar...
 4.6.10: Prove that all solutions of the following secondorder equations ar...
 4.6.11: Prove that all solutions of the following secondorder equations ar...
 4.6.12: Show that all solutions z(t) of  d 2z +z~z~=o dt2 are periodic if...
 4.6.13: (a) Let L=nx max IaJ./axjl, for Ixxol
 4.6.14: Compute the Picard iterates xj(t) of the initialvalue problem d=Ax...
Solutions for Chapter 4.6: Qualitative properties of orbits
Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics  3rd Edition
ISBN: 9780387908069
Solutions for Chapter 4.6: Qualitative properties of orbits
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations and Their Applications: An Introduction to Applied Mathematics, edition: 3. This expansive textbook survival guide covers the following chapters and their solutions. Since 14 problems in chapter 4.6: Qualitative properties of orbits have been answered, more than 6485 students have viewed full stepbystep solutions from this chapter. Chapter 4.6: Qualitative properties of orbits includes 14 full stepbystep solutions. Differential Equations and Their Applications: An Introduction to Applied Mathematics was written by and is associated to the ISBN: 9780387908069.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det 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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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.

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 .

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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