- 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
Tv = Av + Vo = linear transformation plus shift.
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
peA) = det(A - AI) has peA) = zero matrix.
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
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).
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.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
= Xl (column 1) + ... + xn(column n) = combination of columns.
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
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.