 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 13634 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).

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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).

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

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to 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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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.

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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.