 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 5438 students have viewed full stepbystep answer. This expansive textbook survival guide covers the following chapters: 91.

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

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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).

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.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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].

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

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

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.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Outer product uv T
= column times row = rank one matrix.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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

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