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Differential Equations 4th Edition - Solutions by Chapter

Full solutions for Differential Equations | 4th Edition

ISBN: 9780321964670

Differential Equations | 4th Edition - Solutions by Chapter

Solutions by Chapter
4 5 0 259 Reviews
ISBN: 9780321964670

This textbook survival guide was created for the textbook: Differential Equations, edition: 4. Differential Equations was written by and is associated to the ISBN: 9780321964670. The full step-by-step solution to problem in Differential Equations were answered by , our top Math solution expert on 03/13/18, 06:45PM. Since problems from 91 chapters in Differential Equations have been answered, more than 9799 students have viewed full step-by-step answer. This expansive textbook survival guide covers the following chapters: 91.

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).

• Cayley-Hamilton Theorem.

peA) = det(A - AI) has peA) = zero matrix.

• 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.

• Diagonal matrix D.

dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

• Distributive Law

A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

• Echelon matrix U.

The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

• Free columns of A.

Columns without pivots; these are combinations of earlier columns.

• 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.

• Hermitian matrix A H = AT = A.

Complex analog a j i = aU of a symmetric matrix.

• Identity matrix I (or In).

Diagonal entries = 1, off-diagonal entries = 0.

• Iterative method.

A sequence of steps intended to approach the desired solution.

• Kronecker product (tensor product) A ® B.

Blocks aij B, eigenvalues Ap(A)Aq(B).

• Lucas numbers

Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

• Permutation matrix P.

There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

• 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.

• Rotation matrix

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

• Skew-symmetric matrix K.

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

• Special solutions to As = O.

One free variable is Si = 1, other free variables = o.

• Standard basis for Rn.

Columns of n by n identity matrix (written i ,j ,k in R3).

• Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

T- 1 has rank 1 above and below diagonal.

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