 9.6.1: In each of 1 through 4, construct a suitable Liapunov function of t...
 9.6.2: In each of 1 through 4, construct a suitable Liapunov function of t...
 9.6.3: In each of 1 through 4, construct a suitable Liapunov function of t...
 9.6.4: In each of 1 through 4, construct a suitable Liapunov function of t...
 9.6.5: Consider the system of equationsdx/dt = y xf(x, y), dy/dt = x yf(x,...
 9.6.6: A generalization of the undamped pendulum equation isd2u/dt2 + g(u)...
 9.6.7: By introducing suitable dimensionless variables, we can write the s...
 9.6.8: The Linard equation ( of Section 9.3) isd2udt2 + c(u)dudt + g(u) = ...
 9.6.9: (a) A special case of the Linard equation of isd2udt2 +dudt + g(u) ...
 9.6.10: In 10 and 11, we will prove part of Theorem 9.3.2: If the critical ...
 9.6.11: In 10 and 11, we will prove part of Theorem 9.3.2: If the critical ...
 9.6.12: In this problem we prove a part of Theorem 9.3.2 related to instabi...
Solutions for Chapter 9.6: Liapunovs Second Method
Full solutions for Elementary Differential Equations and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 9.6: Liapunovs Second Method
Get Full SolutionsSince 12 problems in chapter 9.6: Liapunovs Second Method have been answered, more than 18228 students have viewed full stepbystep solutions from this chapter. Chapter 9.6: Liapunovs Second Method includes 12 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310. This expansive textbook survival guide covers the following chapters and their solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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!

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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

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

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

Outer product uv T
= column times row = rank one matrix.

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

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 ().

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

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.