- 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: SturmLiouville Boundary Value Problems
- Chapter 11.3: Nonhomogeneous Boundary Value Problems
- Chapter 11.4: Singular SturmLiouville 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 Difference Equations
- Chapter 2.1: Linear Equations; Method of Integrating Factors
- Chapter 2.2: Separable Equations
- Chapter 2.3: Modeling with First Order Equations
- Chapter 2.4: Differences Between Linear and Nonlinear Equations
- Chapter 2.5: Autonomous Equations and Population Dynamics
- Chapter 2.6: Exact 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 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 Vibrations
- Chapter 4.1: General Theory of nth Order Linear Equations
- Chapter 4.2: Homogeneous 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: Review of 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 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 RungeKutta Method
- Chapter 8.4: Multistep Methods
- Chapter 8.5: More on Errors; Stability
- Chapter 8.6: Systems of First Order Equations
- 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: PredatorPrey 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 9th Edition - Solutions by Chapter
Full solutions for Elementary Differential Equations and Boundary Value Problems | 9th Edition
Elementary Differential Equations and Boundary Value Problems | 9th Edition - Solutions by ChapterGet Full Solutions
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Invert A by row operations on [A I] to reach [I A-I].
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.
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 inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
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!
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
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.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
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