- 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
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
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).
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.
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.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + 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.
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.
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).
Constant down each diagonal = time-invariant (shift-invariant) 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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