 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.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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

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

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.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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

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