- Chapter 1: First-order differential equations
- Chapter 1.10: The existence-uniqueness theorem; Picard iteration
- Chapter 1.11: Finding roots of equations by iteration
- Chapter 1.12: Difference equations, and how to compute the interest due on your student loans
- Chapter 1.13: Numerical approximations; Euler's method
- Chapter 1.14: The three term Taylor series method
- Chapter 1.15: An improved Euler method
- Chapter 1.16: The Runge-Kutta method
- Chapter 1.17: What to do in practice
- Chapter 1.2: First-order linear differential equations
- Chapter 1.3: The Van Meegeren art forgeries
- Chapter 1.4: Separable equations
- Chapter 1.5: Population models
- Chapter 1.6: The spread of technological innovations
- Chapter 1.7: An atomic waste disposal problem
- Chapter 1.8: The dynamics of tumor growth, mixing problems and orthogonal trajectories
- Chapter 1.9: Exact equations, and why we cannot solve very many differential equations
- Chapter 2: Second-order linear differential equations
- Chapter 2.1: Algebraic properties of solutions
- Chapter 2.1 1: Differential equations with discontinuous right-hand sides
- Chapter 2.10: Some useful properties of Laplace transforms
- Chapter 2.12: The Dirac delta function
- Chapter 2.13: The convolution integral
- Chapter 2.14: The method of elimination for systems
- Chapter 2.15: Higher-order equations
- Chapter 2.2: Linear equations with constant coefficients
- Chapter 2.3: The nonhomogeneous equation
- Chapter 2.4: The method of variation of parameters
- Chapter 2.5: The method of judicious guessing
- Chapter 2.6: Mechanical vibrations
- Chapter 2.7: A model for the detection of diabetes
- Chapter 2.8: Series solutions
- Chapter 2.9: The method of Laplace transforms
- Chapter 3.1: Algebraic properties of solutions of linear systems
- Chapter 3.1 1: Fundamental matrix solutions; e *'
- Chapter 3.10: Equal roots
- Chapter 3.12: The nonhomogeneous equation; variation of parameters
- Chapter 3.13: Solving systems by Laplace transforms
- Chapter 3.2: Vector spaces
- Chapter 3.3: Dimension of a vector space
- Chapter 3.4: Applications of linear algebra to differential equations
- Chapter 3.5: The theory of determinants
- Chapter 3.6: Solutions of simultaneous linear equations
- Chapter 3.7: Linear transformations
- Chapter 3.8: The eigenvalue-eigenvector method of finding solutions
- Chapter 3.9: Complex roots
- Chapter 4.1: Introduction
- Chapter 4.1 1: The principle of competitive exclusion in population biology
- Chapter 4.12: The Threshold Theorem of epidemiology
- Chapter 4.13: A model for the spread of gonorrhea
- Chapter 4.2: Stability of linear systems
- Chapter 4.3: Stability of equilibrium solutions
- Chapter 4.4: The phase-plane
- Chapter 4.5: Mathematical theories of war
- Chapter 4.6: Qualitative properties of orbits
- Chapter 4.7: Phase portraits of linear systems
- Chapter 4.8: Long time behavior of solutions; the PoincarLBendixson Theorem
- Chapter 4.9: Introduction to bifurcation theory
- Chapter 5.1: Two point boundary-value problems
- Chapter 5.3: Introduction to partial differential equations
- Chapter 5.4: Fourier series
- Chapter 5.5: Even and odd functions
- Chapter 5.6: Return to the heat equation
- Chapter 5.7: The wave equation
- Chapter 5.8: Laplace's equation
Differential Equations and Their Applications: An Introduction to Applied Mathematics 3rd Edition - Solutions by Chapter
Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics | 3rd Edition
Differential Equations and Their Applications: An Introduction to Applied Mathematics | 3rd Edition - Solutions by ChapterGet Full Solutions
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Remove row i and column j; multiply the determinant by (-I)i + j •
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
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.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
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.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
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.
Every v in V is orthogonal to every w in W.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
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.
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