 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
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Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Kirchhoff's Laws.
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.

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.

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 •

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
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