 Chapter 1: FirstOrder Differential Equations
 Chapter 3: Linear Equations of Higher Order
 Chapter 4: Introduction to Systems of Differential Equations
 Chapter 5: Linear Systems of Differential Equations
 Chapter 6: Nonlinear Systems and Phenomena
 Chapter 7: Laplace Transform Methods
Differential Equations: Computing and Modeling 5th Edition  Solutions by Chapter
Full solutions for Differential Equations: Computing and Modeling  5th Edition
ISBN: 9780321816252
Differential Equations: Computing and Modeling  5th Edition  Solutions by Chapter
Get Full SolutionsThe full stepbystep solution to problem in Differential Equations: Computing and Modeling were answered by , our top Math solution expert on 01/24/18, 05:45AM. Differential Equations: Computing and Modeling was written by and is associated to the ISBN: 9780321816252. This expansive textbook survival guide covers the following chapters: 6. Since problems from 6 chapters in Differential Equations: Computing and Modeling have been answered, more than 2503 students have viewed full stepbystep answer. This textbook survival guide was created for the textbook: Differential Equations: Computing and Modeling, edition: 5.

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

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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

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.

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

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Solvable system Ax = b.
The right side b is in the column space of A.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

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

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.

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