- Chapter 1.1: Some Basic Mathematical Models; Direction Fields
- Chapter 1.2: Solutions of Some Differential Equations
- Chapter 1.3: Classification of Differential Equations
- Chapter 10.1: Two-Point Boundary Value Problems
- Chapter 10.2: Fourier Series
- Chapter 10.3: The Fourier Convergence Theorem
- Chapter 10.4: Even and Odd Functions
- Chapter 10.5: Separation of Variables; Heat Conduction in a Rod
- Chapter 10.6: Other Heat Conduction Problems
- Chapter 10.7: The Wave Equation: Vibrations of an Elastic String
- Chapter 10.8: Laplaces Equation
- Chapter 11.1: The Occurrence of Two-Point Boundary Value Problems
- Chapter 11.2: Sturm-Liouville Boundary Value Problems
- Chapter 11.3: Nonhomogeneous Boundary Value Problems
- Chapter 11.4: Singular Sturm-Liouville Problems
- Chapter 11.5: Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion
- Chapter 11.6: Series of Orthogonal Functions: Mean Convergence
- Chapter 2: First-Order Differential Equations
- Chapter 2.1: Linear Differential Equations; Method of Integrating Factors
- Chapter 2.2: Separable Differential Equations
- Chapter 2.3: Modeling with First-Order Differential Equations
- Chapter 2.4: Differences Between Linear and Nonlinear Differential Equations
- Chapter 2.5: Autonomous Differential Equations and Population Dynamics
- Chapter 2.6: Exact Differential Equations and Integrating Factors
- Chapter 2.7: Numerical Approximations: Eulers Method
- Chapter 2.8: The Existence and Uniqueness Theorem
- Chapter 2.9: First-Order Difference Equations
- Chapter 3.1: Homogeneous Differential Equations with Constant Coefficients
- Chapter 3.2: Solutions of Linear Homogeneous Equations; the Wronskian
- Chapter 3.3: Complex Roots of the Characteristic Equation
- Chapter 3.4: Repeated Roots; Reduction of Order
- Chapter 3.5: Nonhomogeneous Equations; Method of Undetermined Coefficients
- Chapter 3.6: Variation of Parameters
- Chapter 3.7: Mechanical and Electrical Vibrations
- Chapter 3.8: Forced Periodic Vibrations
- Chapter 4.1: General Theory of nth Order
- Chapter 4.2: Homogeneous Differential Equations with Constant Coefficients
- Chapter 4.3: The Method of Undetermined Coefficients
- Chapter 4.4: The Method of Variation of Parameters
- Chapter 5.1: Review of Power Series
- Chapter 5.2: Series Solutions Near an Ordinary Point, Part I
- Chapter 5.3: Series Solutions Near an Ordinary Point, Part II
- Chapter 5.4: Euler Equations; Regular Singular Points
- Chapter 5.5: Series Solutions Near a Regular Singular Point, Part I
- Chapter 5.6: Series Solutions Near a Regular Singular Point, Part II
- Chapter 5.7: Bessels Equation
- Chapter 6.1: Definition of the Laplace Transform
- Chapter 6.2: Solution of Initial Value Problems
- Chapter 6.3: Step Functions
- Chapter 6.4: Differential Equations with Discontinuous Forcing Functions
- Chapter 6.5: Impulse Functions
- Chapter 6.6: The Convolution Integral
- Chapter 7.1: Introduction
- Chapter 7.2: Matrices
- Chapter 7.3: Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors
- Chapter 7.4: Basic Theory of Systems of First-Order Linear Equations
- Chapter 7.5: Homogeneous Linear Systems with Constant Coefficients
- Chapter 7.6: Complex-Valued Eigenvalues
- Chapter 7.7: Fundamental Matrices
- Chapter 7.8: Repeated Eigenvalues
- Chapter 7.9: Nonhomogeneous Linear Systems
- Chapter 8.1: The Euler or Tangent Line Method
- Chapter 8.2: Improvements on the Euler Method
- Chapter 8.3: The Runge-Kutta Method
- Chapter 8.4: Multistep Methods
- Chapter 8.5: Systems of First-Order Equations
- Chapter 8.6: More on Errors; Stability
- Chapter 9.1: The Phase Plane: Linear Systems
- Chapter 9.2: Autonomous Systems and Stability
- Chapter 9.3: Locally Linear Systems
- Chapter 9.4: Competing Species
- Chapter 9.5: Predator -- Prey Equations
- Chapter 9.6: Liapunovs Second Method
- Chapter 9.7: Periodic Solutions and Limit Cycles
- Chapter 9.8: Chaos and Strange Attractors: The Lorenz Equations
Elementary Differential Equations and Boundary Value Problems 11th Edition - Solutions by Chapter
Full solutions for Elementary Differential Equations and Boundary Value Problems | 11th Edition
Elementary Differential Equations and Boundary Value Problems | 11th Edition - Solutions by ChapterGet Full Solutions
Tv = Av + Vo = linear transformation plus shift.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
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).
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
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.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.