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# Solutions for Chapter 3.6: SECOND-ORDER LINEAR EQUATIONS

## Full solutions for Differential Equations 00 | 4th Edition

ISBN: 9780495561989

Solutions for Chapter 3.6: SECOND-ORDER LINEAR EQUATIONS

Solutions for Chapter 3.6
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##### ISBN: 9780495561989

This textbook survival guide was created for the textbook: Differential Equations 00, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 3.6: SECOND-ORDER LINEAR EQUATIONS includes 40 full step-by-step solutions. Since 40 problems in chapter 3.6: SECOND-ORDER LINEAR EQUATIONS have been answered, more than 16069 students have viewed full step-by-step solutions from this chapter. Differential Equations 00 was written by and is associated to the ISBN: 9780495561989.

Key Math Terms and definitions covered in this textbook
• Adjacency matrix of a graph.

Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

• Cholesky factorization

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

• Column space C (A) =

space of all combinations of the columns of A.

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.

• 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.

• Dimension of vector space

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

• Elimination.

A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

• Factorization

A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

• 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.

• Krylov subspace Kj(A, b).

The subspace spanned by b, Ab, ... , Aj-Ib. 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.

• lA-II = 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.

• Norm

IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

• Partial pivoting.

In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

• Plane (or hyperplane) in Rn.

Vectors x with aT x = O. Plane is perpendicular to a =1= O.

• Reflection matrix (Householder) Q = I -2uuT.

Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• Skew-symmetric 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 = time-invariant (shift-invariant) filter.

• Volume of box.

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

• Wavelets Wjk(t).

Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).

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