 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 9799 students have viewed full stepbystep answer. This expansive textbook survival guide covers the following chapters: 91.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Iterative method.
A sequence of steps intended to approach the desired solution.

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

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.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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.

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 ().

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.