- Chapter 1.1:
- Chapter 1.1: DEFINITIONS AND TERMINOLOGY
- Chapter 1.2:
- Chapter 1.2: INITIAL-VALUE 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 FIRST-ORDER 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: CAUCHY-EULER 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: INITIAL-VALUE PROBLEMS
- Chapter 5.2:
- Chapter 5.2: LINEAR MODELS: BOUNDARY-VALUE 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: RUNGE-KUTTA METHODS
- Chapter 9.3:
- Chapter 9.3: MULTISTEP METHODS
- Chapter 9.4:
- Chapter 9.4: HIGHER-ORDER EQUATIONS AND SYSTEMS
- Chapter 9.5:
- Chapter 9.5: SECOND-ORDER BOUNDARY-VALUE 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: FIRST-ORDER DIFFERENTIAL EQUATIONS
- Chapter Chapter 3: MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS
- Chapter Chapter 4: HIGHER-ORDER DIFFERENTIAL EQUATIONS
- Chapter Chapter 5: MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS
- Chapter Chapter 6: SERIES SOLUTIONS OF LINEAR EQUATIONS
- Chapter Chapter 7: SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS
- Chapter Chapter 8: SYSTEMS OF LINEAR FIRST-ORDER 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
A First Course in Differential Equations with Modeling Applications | 10th Edition - Solutions by ChapterGet Full Solutions
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
A = CTC = (L.J]))(L.J]))T for positive definite A.
Column space C (A) =
space of all combinations of the columns of A.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
lA-II = 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.
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).
A directed graph that has constants Cl, ... , Cm associated with the edges.
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.
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.
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.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Having trouble accessing your account? Let us help you, contact support at +1(510) 944-1054 or firstname.lastname@example.org
Forgot password? Reset it here