 Chapter 1.1:
 Chapter 1.1: DEFINITIONS AND TERMINOLOGY
 Chapter 1.2:
 Chapter 1.2: INITIALVALUE PROBLEMS
 Chapter 1.3:
 Chapter 1.3: DIFFERENTIAL EQUATIONS AS MATHEMATICAL MODEL
 Chapter 1.R:
 Chapter 2.1:
 Chapter 2.1: SOLUTION CURVES WITHOUT A SOLUTION
 Chapter 2.2:
 Chapter 2.2: SEPARABLE EQUATIONS
 Chapter 2.3:
 Chapter 2.3: LINEAR EQUATIONS
 Chapter 2.4:
 Chapter 2.4: EXACT EQUATIONS
 Chapter 2.5:
 Chapter 2.5: SOLUTIONS BY SUBSTITUTIONS
 Chapter 2.6:
 Chapter 2.6: A NUMERICAL METHOD
 Chapter 2.R:
 Chapter 3.1:
 Chapter 3.1: LINEAR MODELS
 Chapter 3.2:
 Chapter 3.2: NONLINEAR MODELS
 Chapter 3.3:
 Chapter 3.3: MODELING WITH SYSTEMS OF FIRSTORDER DEs
 Chapter 3.R:
 Chapter 4.1:
 Chapter 4.1: PRELIMINARY THEORYLINEAR EQUATIONS
 Chapter 4.10:
 Chapter 4.2:
 Chapter 4.2: REDUCTION OF ORDER
 Chapter 4.3:
 Chapter 4.3: HOMOGENEOUS LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS
 Chapter 4.4:
 Chapter 4.4: UNDETERMINED COEFFICIENTSSUPERPOSITION APPROACH
 Chapter 4.5:
 Chapter 4.5: UNDETERMINED COEFFICIENTSANNIHILATOR APPROACH
 Chapter 4.6:
 Chapter 4.6: VARIATION OF PARAMETERS
 Chapter 4.7:
 Chapter 4.7: CAUCHYEULER EQUATION
 Chapter 4.8:
 Chapter 4.8: GREENS FUNCTIONS
 Chapter 4.9:
 Chapter 4.9: SOLVING SYSTEMS OF LINEAR DES BY ELIMINATION
 Chapter 4.R:
 Chapter 5.1:
 Chapter 5.1: LINEAR MODELS: INITIALVALUE PROBLEMS
 Chapter 5.2:
 Chapter 5.2: LINEAR MODELS: BOUNDARYVALUE PROBLEMS
 Chapter 5.3:
 Chapter 5.3: NONLINEAR MODEL
 Chapter 5.R:
 Chapter 6.1:
 Chapter 6.1: REVIEW OF POWER SERIES
 Chapter 6.2:
 Chapter 6.2: SOLUTIONS ABOUT ORDINARY POINTS
 Chapter 6.3:
 Chapter 6.3: SOLUTIONS ABOUT SINGULAR POINTS
 Chapter 6.4:
 Chapter 6.4: SPECIAL FUNCTIONS
 Chapter 6.R:
 Chapter 7.1:
 Chapter 7.1: DEFINITION OF THE LAPLACE TRANSFORM
 Chapter 7.2:
 Chapter 7.2: INVERSE TRANSFORMS AND TRANSFORMS OF DERIVATIVES
 Chapter 7.3:
 Chapter 7.3: OPERATIONAL PROPERTIES
 Chapter 7.4:
 Chapter 7.4: OPERATIONAL PROPERTIES II
 Chapter 7.5:
 Chapter 7.5: THE DIRAC DELTA FUNCTION
 Chapter 7.6:
 Chapter 7.6: SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS
 Chapter 7.R:
 Chapter 8.1:
 Chapter 8.1: PRELIMINARY THEORYLINEAR SYSTEMS
 Chapter 8.2:
 Chapter 8.2: HOMOGENEOUS LINEAR SYSTEMS
 Chapter 8.3:
 Chapter 8.3: NONHOMOGENEOUS LINEAR SYSTEMS
 Chapter 8.4:
 Chapter 8.4: MATRIX EXPONENTIAL
 Chapter 8.R:
 Chapter 9.1:
 Chapter 9.1: EULER METHODS AND ERROR ANALYSIS
 Chapter 9.2:
 Chapter 9.2: RUNGEKUTTA METHODS
 Chapter 9.3:
 Chapter 9.3: MULTISTEP METHODS
 Chapter 9.4:
 Chapter 9.4: HIGHERORDER EQUATIONS AND SYSTEMS
 Chapter 9.5:
 Chapter 9.5: SECONDORDER BOUNDARYVALUE PROBLEMS
 Chapter 9.R:
 Chapter A.I:
 Chapter A.II:
 Chapter APPENDIX I: GAMMA FUNCTION
 Chapter APPENDIX II: MATRICES
 Chapter Chapter 1: INTRODUCTION TO DIFFERENTIAL EQUATIONS
 Chapter Chapter 2: FIRSTORDER DIFFERENTIAL EQUATIONS
 Chapter Chapter 3: MODELING WITH FIRSTORDER DIFFERENTIAL EQUATIONS
 Chapter Chapter 4: HIGHERORDER DIFFERENTIAL EQUATIONS
 Chapter Chapter 5: MODELING WITH HIGHERORDER DIFFERENTIAL EQUATIONS
 Chapter Chapter 6: SERIES SOLUTIONS OF LINEAR EQUATIONS
 Chapter Chapter 7: SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS
 Chapter Chapter 8: SYSTEMS OF LINEAR FIRSTORDER DIFFERENTIAL EQUATIONS
 Chapter Chapter 9: NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
A First Course in Differential Equations with Modeling Applications 10th Edition  Solutions by Chapter
Full solutions for A First Course in Differential Equations with Modeling Applications  10th Edition
ISBN: 9781111827052
A First Course in Differential Equations with Modeling Applications  10th Edition  Solutions by Chapter
Get Full SolutionsThis textbook survival guide was created for the textbook: A First Course in Differential Equations with Modeling Applications, edition: 10th. This expansive textbook survival guide covers the following chapters: 109. A First Course in Differential Equations with Modeling Applications was written by and is associated to the ISBN: 9781111827052. The full stepbystep solution to problem in A First Course in Differential Equations with Modeling Applications were answered by , our top Math solution expert on 07/17/17, 09:41AM. Since problems from 109 chapters in A First Course in Differential Equations with Modeling Applications have been answered, more than 22647 students have viewed full stepbystep answer.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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].

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= 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.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Stiffness 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.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).