 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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Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Column space C (A) =
space of all combinations of the columns of A.

Condition number
cond(A) = c(A) = IIAIlIIAIII = 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.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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 .

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).