 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: TwoPoint 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 TwoPoint 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
ISBN: 9780470383346
Elementary Differential Equations and Boundary Value Problems  9th Edition  Solutions by Chapter
Get Full SolutionsSince problems from 75 chapters in Elementary Differential Equations and Boundary Value Problems have been answered, more than 3975 students have viewed full stepbystep answer. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9. This expansive textbook survival guide covers the following chapters: 75. The full stepbystep solution to problem in Elementary Differential Equations and Boundary Value Problems were answered by , our top Math solution expert on 03/13/18, 08:22PM.

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).

Condition number
cond(A) = c(A) = IIAIlIIAIII = 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. Blockdiagonal: 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 IIA1 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.

Multiplier eij.
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 •

Pascal matrix
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.

Schwarz inequality
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.

Skewsymmetric 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.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
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