 4.4.1: y x y = 1 x
 4.4.11: In each of 1 through 10, find the first five terms of each of two l...
 4.4.2: y x 3 y = 4
 4.4.12: In each of 1 through 10, find the first five terms of each of two l...
 4.4.3: y+ (1 x 2 )y = x
 4.4.13: In each of 1 through 10, find the first five terms of each of two l...
 4.4.4: y+ 2y+ x y = 0 5
 4.4.14: In each of 1 through 10, find the first five terms of each of two l...
 4.4.5: y x y+ y = 3 6
 4.4.15: In each of 1 through 10, find the first five terms of each of two l...
 4.4.6: y+ x y+ x y = 0 7
 4.4.16: In each of 1 through 10, find the first five terms of each of two l...
 4.4.7: y x 2 y+ 2y = x 8
 4.4.17: In each of 1 through 10, find the first five terms of each of two l...
 4.4.8: y+ x y = cos(x)
 4.4.18: In each of 1 through 10, find the first five terms of each of two l...
 4.4.9: y+ (1 x)y+ 2y = 1 x 2 1
 4.4.19: In each of 1 through 10, find the first five terms of each of two l...
 4.4.10: y+ x y= 1 ex 4
 4.4.20: In each of 1 through 10, find the first five terms of each of two l...
Solutions for Chapter 4: Series Solutions
Full solutions for Advanced Engineering Mathematics  7th Edition
ISBN: 9781111427412
Solutions for Chapter 4: Series Solutions
Get Full SolutionsAdvanced Engineering Mathematics was written by and is associated to the ISBN: 9781111427412. Chapter 4: Series Solutions includes 20 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 7. Since 20 problems in chapter 4: Series Solutions have been answered, more than 7773 students have viewed full stepbystep solutions from this chapter.

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.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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

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].

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.

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

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

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.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

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

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

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·

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.