- 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).
Remove row i and column j; multiply the determinant by (-I)i + j •
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).
Dimension of vector space
dim(V) = number of vectors in any basis for V.
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.
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.
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+ (Moore-Penrose 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 = Q-1 = 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)j-I and det V = product of (Xk - Xi) for k > i.