- Chapter 1.1: Differential Equations Everywhere
- Chapter 1.10: Numerical Solution to First-Order Differential Equations
- Chapter 1.11: Some Higher-Order Differential Equations
- Chapter 1.12: Basic Theory of Differential Equations
- Chapter 1.2: Basic Ideas and Terminology
- Chapter 1.3: The Geometry of First-Order Differential Equations
- Chapter 1.4: Separable Differential Equations
- Chapter 1.5: Some Simple Population Models
- Chapter 1.6: First-Order Linear Differential Equations
- Chapter 1.7: Modeling Problems Using First-Order 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 Initial-Value 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: Row-Echelon 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 Rank-Nullity 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: Complex-Valued 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: First-Order 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 First-Order Linear Differential Systems
- Chapter 9.4: Vector Differential Equations: Nondefective Coefficient Matrix
- Chapter 9.5: Vector Differential Equations: Defective Coefficient Matrix
- Chapter 9.6: Variation-of-Parameters 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
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
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 n-l - An).
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
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.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Gram-Schmidt 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.
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.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
A directed graph that has constants Cl, ... , Cm associated with the edges.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.