- 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
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).
peA) = det(A - AI) has peA) = zero matrix.
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. Block-diagonal: zero outside square blocks Du.
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.
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, off-diagonal entries = 0.
A sequence of steps intended to approach the desired solution.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Skew-symmetric 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.