 Chapter 1: Firstorder differential equations
 Chapter 1.10: The existenceuniqueness 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 RungeKutta method
 Chapter 1.17: What to do in practice
 Chapter 1.2: Firstorder 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: Secondorder linear differential equations
 Chapter 2.1: Algebraic properties of solutions
 Chapter 2.1 1: Differential equations with discontinuous righthand 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: Higherorder 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 eigenvalueeigenvector 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 phaseplane
 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 boundaryvalue 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
ISBN: 9780387908069
Differential Equations and Their Applications: An Introduction to Applied Mathematics  3rd Edition  Solutions by Chapter
Get Full SolutionsSince problems from 65 chapters in Differential Equations and Their Applications: An Introduction to Applied Mathematics have been answered, more than 1866 students have viewed full stepbystep answer. This textbook survival guide was created for the textbook: Differential Equations and Their Applications: An Introduction to Applied Mathematics, edition: 3. This expansive textbook survival guide covers the following chapters: 65. The full stepbystep solution to problem in Differential Equations and Their Applications: An Introduction to Applied Mathematics were answered by Patricia, our top Math solution expert on 03/13/18, 07:00PM. Differential Equations and Their Applications: An Introduction to Applied Mathematics was written by Patricia and is associated to the ISBN: 9780387908069.

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.

Cofactor Cij.
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.

Covariance matrix:E.
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.

Cyclic shift
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.

Elimination.
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 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.

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 Fn1c 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 .

Norm
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 subspaces.
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.

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.
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