- 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
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).
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
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
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
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.
Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
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.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
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.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
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