 Chapter 1: INTRODUCTION TO DIFFERENTIAL EQUATIONS
 Chapter 1.1: INTRODUCTION TO DIFFERENTIAL EQUATIONS
 Chapter 1.2: INTRODUCTION TO DIFFERENTIAL EQUATIONS
 Chapter 1.3: INTRODUCTION TO DIFFERENTIAL EQUATIONS
 Chapter 10: Plane Autonomous Systems
 Chapter 10.1: Plane Autonomous Systems
 Chapter 10.2: Plane Autonomous Systems
 Chapter 10.3: Plane Autonomous Systems
 Chapter 10.4: Plane Autonomous Systems
 Chapter 11: Fourier Series
 Chapter 11.1: Fourier Series
 Chapter 11.2: Fourier Series
 Chapter 11.3: Fourier Series
 Chapter 11.4: Fourier Series
 Chapter 11.5: Fourier Series
 Chapter 12: BoundaryValue Problems in Rectangular Coordinates
 Chapter 12.1: BoundaryValue Problems in Rectangular Coordinates
 Chapter 12.2: BoundaryValue Problems in Rectangular Coordinates
 Chapter 12.3: BoundaryValue Problems in Rectangular Coordinates
 Chapter 12.4: BoundaryValue Problems in Rectangular Coordinates
 Chapter 12.5: BoundaryValue Problems in Rectangular Coordinates
 Chapter 12.6: BoundaryValue Problems in Rectangular Coordinates
 Chapter 12.7: BoundaryValue Problems in Rectangular Coordinates
 Chapter 12.8: BoundaryValue Problems in Rectangular Coordinates
 Chapter 13: BoundaryValue Problems in Other Coordinate Systems
 Chapter 13.1: BoundaryValue Problems in Other Coordinate Systems
 Chapter 13.2: BoundaryValue Problems in Other Coordinate Systems
 Chapter 13.3: BoundaryValue Problems in Other Coordinate Systems
 Chapter 14.1: Integral Transforms
 Chapter 14.2: Integral Transforms
 Chapter 14.3: Integral Transforms
 Chapter 14.4: Integral Transforms
 Chapter 14.5: Integral Transforms
 Chapter 15: Numerical Solutions of Partial Differential Equations
 Chapter 15.1: Numerical Solutions of Partial Differential Equations
 Chapter 15.2: Numerical Solutions of Partial Differential Equations
 Chapter 15.3: Numerical Solutions of Partial Differential Equations
 Chapter 2: FirstOrder Differential Equations
 Chapter 2.1: FirstOrder Differential Equations
 Chapter 2.2: FirstOrder Differential Equations
 Chapter 2.3: FirstOrder Differential Equations
 Chapter 2.4: FirstOrder Differential Equations
 Chapter 2.5: FirstOrder Differential Equations
 Chapter 2.6: FirstOrder Differential Equations
 Chapter 3: Modeling with FirstOrder Differential Equations
 Chapter 3.1: Modeling with FirstOrder Differential Equations
 Chapter 3.2: Modeling with FirstOrder Differential Equations
 Chapter 3.3: Modeling with FirstOrder Differential Equations
 Chapter 4: HigherOrder Differential Equations
 Chapter 4.1: HigherOrder Differential Equations
 Chapter 4.2: HigherOrder Differential Equations
 Chapter 4.3: HigherOrder Differential Equations
 Chapter 4.4: HigherOrder Differential Equations
 Chapter 4.5: HigherOrder Differential Equations
 Chapter 4.6: HigherOrder Differential Equations
 Chapter 4.7: HigherOrder Differential Equations
 Chapter 4.8: HigherOrder Differential Equations
 Chapter 4.9: HigherOrder Differential Equations
 Chapter 5: Modeling with HigherOrder Differential Equations
 Chapter 5.1: Modeling with HigherOrder Differential Equations
 Chapter 5.2: Modeling with HigherOrder Differential Equations
 Chapter 5.3: Modeling with HigherOrder Differential Equations
 Chapter 6: Series Solutions of Linear Equations
 Chapter 6.1: Series Solutions of Linear Equations
 Chapter 6.2: Series Solutions of Linear Equations
 Chapter 6.3: Series Solutions of Linear Equations
 Chapter 6.4: Series Solutions of Linear Equations
 Chapter 7: The Laplace Transform
 Chapter 7.1: The Laplace Transform
 Chapter 7.2: The Laplace Transform
 Chapter 7.3: The Laplace Transform
 Chapter 7.4: The Laplace Transform
 Chapter 7.5: The Laplace Transform
 Chapter 7.6: The Laplace Transform
 Chapter 8: Systems of Linear FirstOrder Differential Equations
 Chapter 8.1: Systems of Linear FirstOrder Differential Equations
 Chapter 8.2: Systems of Linear FirstOrder Differential Equations
 Chapter 8.3: Systems of Linear FirstOrder Differential Equations
 Chapter 8.4: Systems of Linear FirstOrder Differential Equations
 Chapter 9: Numerical Solutions of Ordinary Differential Equations
 Chapter 9.1: Numerical Solutions of Ordinary Differential Equations
 Chapter 9.2: Numerical Solutions of Ordinary Differential Equations
 Chapter 9.3: Numerical Solutions of Ordinary Differential Equations
 Chapter 9.4: Numerical Solutions of Ordinary Differential Equations
 Chapter 9.5: Numerical Solutions of Ordinary Differential Equations
Differential Equations with BoundaryValue Problems, 8th Edition  Solutions by Chapter
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Differential Equations with BoundaryValue Problems,  8th Edition  Solutions by Chapter
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Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

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

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. 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.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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).

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

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