- 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
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
peA) = det(A - AI) has peA) = zero matrix.
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.
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
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.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
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.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
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