 Chapter 1.1: Differential Equations Everywhere
 Chapter 1.10: Numerical Solution to FirstOrder Differential Equations
 Chapter 1.11: Some HigherOrder Differential Equations
 Chapter 1.12: Basic Theory of Differential Equations
 Chapter 1.2: Basic Ideas and Terminology
 Chapter 1.3: The Geometry of FirstOrder Differential Equations
 Chapter 1.4: Separable Differential Equations
 Chapter 1.5: Some Simple Population Models
 Chapter 1.6: FirstOrder Linear Differential Equations
 Chapter 1.7: Modeling Problems Using FirstOrder Linear Differential Equations
 Chapter 1.8: Change of Variables
 Chapter 1.9: Exact Differential Equations
 Chapter 10.1: Definition of the Laplace Transform
 Chapter 10.10: The Laplace Transform and Some Elementary Applications
 Chapter 10.2: The Existence of the Laplace Transform and the Inverse Transform
 Chapter 10.3: Periodic Functions and the Laplace Transform
 Chapter 10.4: The Transform of Derivatives and Solution of InitialValue Problems
 Chapter 10.5: The First Shifting Theorem
 Chapter 10.6: The Unit Step Function
 Chapter 10.7: The Second Shifting Theorem
 Chapter 10.8: Impulsive Driving Terms: The Dirac Delta Function
 Chapter 10.9: The Convolution Integral
 Chapter 11.1: Review of Power Series
 Chapter 11.2: Series Solutions about an Ordinary Point
 Chapter 11.3: The Legendre Equation
 Chapter 11.4: Series Solutions about a Regular Singular Point
 Chapter 11.5: Frobenius Theory
 Chapter 11.6: Bessels Equation of Order p
 Chapter 11.7: Series Solutions to Linear Differential Equations
 Chapter 2.1: Matrices: Definitions and Notation
 Chapter 2.2: Matrix Algebra
 Chapter 2.3: Terminology for Systems of Linear Equations
 Chapter 2.4: RowEchelon Matrices and Elementary Row Operations
 Chapter 2.5: Gaussian Elimination
 Chapter 2.6: The Inverse of a Square Matrix
 Chapter 2.7: Elementary Matrices and the LU Factorization
 Chapter 2.8: The Invertible Matrix Theorem I I
 Chapter 2.9: Chapter Review
 Chapter 3.1: The Definition of the Determinant
 Chapter 3.2: Properties of Determinants F
 Chapter 3.3: Cofactor Expansions
 Chapter 3.4: Summary of Determinants
 Chapter 3.5: Chapter Review
 Chapter 4: Vector Spaces
 Chapter 4.1: Vectors in Rn
 Chapter 4.2: Definition of a Vector Space
 Chapter 4.3: Subspaces
 Chapter 4.4: Spanning Sets
 Chapter 4.5: Linear Dependence and Linear Independence
 Chapter 4.6: Bases and Dimension
 Chapter 4.7: Change of Basis
 Chapter 4.8: Row Space and Column Space
 Chapter 4.9: The RankNullity Theorem
 Chapter 5.1: Definition of an Inner Product Space
 Chapter 5.2: Orthogonal Sets of Vectors and Orthogonal Projections
 Chapter 5.4: Least Squares Approximation
 Chapter 5.5: Inner Product Spaces
 Chapter 6.1: Definition of a Linear Transformation
 Chapter 6.2: Transformations of R2
 Chapter 6.3: The Kernel and Range of a Linear Transformation
 Chapter 6.4: Additional Properties of Linear Transformations
 Chapter 6.5: The Matrix of a Linear Transformation
 Chapter 6.6: Linear Transformations
 Chapter 7.1: The Eigenvalue/Eigenvector Problem
 Chapter 7.2: General Results for Eigenvalues and Eigenvectors
 Chapter 7.3: Diagonalization
 Chapter 7.4: An Introduction to the Matrix Exponential Function
 Chapter 7.5: Orthogonal Diagonalization and Quadratic Forms
 Chapter 7.6: Jordan Canonical Forms
 Chapter 7.7: The Algebraic Eigenvalue/Eigenvector Problem
 Chapter 8.1: General Theory for Linear Differential Equations
 Chapter 8.10: Linear Differential Equations of Order n
 Chapter 8.2: Constant Coefficient Homogeneous Linear Differential Equations
 Chapter 8.3: The Method of Undetermined Coefficients: Annihilators
 Chapter 8.4: ComplexValued Trial Solutions
 Chapter 8.5: Oscillations of a Mechanical System
 Chapter 8.6: RLC Circuits
 Chapter 8.7: The Variation of Parameters Method
 Chapter 8.8: A Differential Equation with Nonconstant Coefficients
 Chapter 8.9: Reduction of Order
 Chapter 9.1: FirstOrder Linear Systems
 Chapter 9.10: Nonlinear Systems
 Chapter 9.11: Systems of Differential Equations
 Chapter 9.2: Vector Formulation
 Chapter 9.3: General Results for FirstOrder Linear Differential Systems
 Chapter 9.4: Vector Differential Equations: Nondefective Coefficient Matrix
 Chapter 9.5: Vector Differential Equations: Defective Coefficient Matrix
 Chapter 9.6: VariationofParameters for Linear Systems
 Chapter 9.7: Some Applications of Linear Systems of Differential Equations
 Chapter 9.8: Matrix Exponential Function and Systems of Differential Equations
 Chapter 9.9: The Phase Plane for Linear Autonomous Systems
Differential Equations 4th Edition  Solutions by Chapter
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Differential Equations  4th Edition  Solutions by Chapter
Get Full SolutionsThis 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 stepbystep 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 32989 students have viewed full stepbystep answer. This expansive textbook survival guide covers the following chapters: 91.

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

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

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

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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).

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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

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

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).