 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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Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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).

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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

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.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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).

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.