 12.1: In 16, find all the critical points for the given system, discuss t...
 12.2: In 16, find all the critical points for the given system, discuss t...
 12.3: In 16, find all the critical points for the given system, discuss t...
 12.4: In 16, find all the critical points for the given system, discuss t...
 12.5: In 16, find all the critical points for the given system, discuss t...
 12.6: In 16, find all the critical points for the given system, discuss t...
 12.7: In 7 and 8, use the potential plane to help sketch the phase plane ...
 12.8: In 7 and 8, use the potential plane to help sketch the phase plane ...
 12.9: In 912, use Lyapunovs direct method to determine the stability for ...
 12.10: In 912, use Lyapunovs direct method to determine the stability for ...
 12.11: In 912, use Lyapunovs direct method to determine the stability for ...
 12.12: In 912, use Lyapunovs direct method to determine the stability for ...
 12.13: In 13 and 14, sketch the phase plane diagrams for the given system....
 12.14: In 13 and 14, sketch the phase plane diagrams for the given system....
 12.15: In 15 and 16, determine whether the given system has a nonconstant ...
 12.16: In 15 and 16, determine whether the given system has a nonconstant ...
 12.17: In 17 and 18, determine the stability of the zero solution to the g...
 12.18: In 17 and 18, determine the stability of the zero solution to the g...
Solutions for Chapter 12: Fundamentals of Differential Equations and Boundary Value Problems 6th Edition
Full solutions for Fundamentals of Differential Equations and Boundary Value Problems  6th Edition
ISBN: 9780321747747
Solutions for Chapter 12
Get Full SolutionsChapter 12 includes 18 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Fundamentals of Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780321747747. Since 18 problems in chapter 12 have been answered, more than 1828 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Fundamentals of Differential Equations and Boundary Value Problems, edition: 6.

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

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.