Differential Equations and Linear Algebra

Undergraduate Mathematics
Science and Engineering
August 2022



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PDF textbook and exercise solutions deBookAndSolutionsGG.pdf (13MB)

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PDF textbook only, no solutions deBookGG.pdf (9MB)
PDF solution manual only, no textbook deSolutionsGG.pdf (6MB)

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Chapter PDF files (0.5 MB to 2.2 MB)

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1 Fundamentals    1
1.1 Exponential Modeling      2
1.2 Exponential Application Library     16
1.3 Differential Equations of First Order 31
1.4 Direction Fields       40
1.5 Phase Line Diagrams       51
1.6 Computing and Existence      64

2 First Order Differential Equations    74
2.1 Quadrature Method      74
2.2 Separable Equations       82
2.3 Linear Equations       93
2.4 Undetermined Coefficients      104
2.5 Linear Applications       111
2.6 Kinetics       125
2.7 Logistic Equation       142
2.8 Science and Engineering Applications    147
2.9 Exact Equations and Level Curves      162
2.10 Special equations    167

3 Linear Algebraic Equations No Matrices    174
3.1 Systems of Linear Equations       174
3.2 Filmstrips and Toolkit Sequences      185
3.3 General Solution Theory      194
3.4 Basis, Dimension, Nullity and Rank    207
3.5 Answer Check, Proofs and Details     218

4 Numerical Methods with Applications    225
4.1 Solving y′ = F(x) Numerically  225
4.2 Solving y′ = f(x, y) Numerically   238
4.3 Error in Numerical Methods       247
4.4 Computing π, ln 2 and e     254
4.5 Earth to the Moon       260
4.6 Skydiving       267
4.7 Lunar Lander        272
4.8 Comets      277
4.9 Fish Farming        284

5 Linear Algebra    293
5.1 Vectors and Matrices       294
5.2 Matrix Equations       321
5.3 Determinants and Cramer’s Rule    343
5.4 Vector Spaces, Independence, Basis      369
5.5 Basis, Dimension and Rank       405

6 Scalar Linear Differential Equations    430
6.1 Linear 2nd Order Constant      430
6.2 Continuous Coefficient Theory     442
6.3 Higher Order Linear Constant Equations     451
6.4 Variation of Parameters      464
6.5 Undetermined Coefficients      470
6.6 Undamped Mechanical Vibrations     490
6.7 Forced and Damped Vibrations      506
6.8 Resonance       528
6.9 Kepler’s laws      546

7 Topics in Linear Differential Equations    551
7.1 Higher Order Homogeneous       551
7.2 Differential Operators     557
7.3 Higher Order Non-Homogeneous     560
7.4 Cauchy-Euler Equation     566
7.5 Variation of Parameters Revisited      569
7.6 Undetermined Coefficients Library      574

8 Laplace Transform    591
8.1 Laplace Method Introduction      592
8.2 Laplace Integral Table      601
8.3 Laplace Transform Rules      609
8.4 Heaviside’s Method for Partial Fractions   620
8.5 Transform Properties       637
8.6 Heaviside Step and Dirac Impulse      644
8.7 Laplace Table Derivations      650
8.8 Modeling    654

9 Eigenanalysis    667
9.1 Matrix Eigenanalysis       667
9.2 Eigenanalysis Applications      703
9.3 Advanced Topics in Linear Algebra      720

10 Phase Plane Methods     750
10.1 Planar Autonomous Systems     751
10.2 Planar Constant Linear Systems     767
10.3 Planar Almost Linear Systems     780
10.4 Biological Models     790
10.5 Mechanical Models     804

11 Systems of Differential Equations     812
11.1 Examples of Systems        813
11.2 Fundamental System Methods       839
11.3 Structure of Linear Systems       850
11.4 Matrix Exponential        865
11.5 Eigenanalysis, Spectral and CHZ Methods     875
11.6 Jordan Form and Eigenanalysis       894
11.7 Nonhomogeneous Linear Systems       912 
11.8 Second-order Systems     923 
11.9 Numerical Methods for Systems     942 

12 Series Methods     949 
12.1 Review of Calculus Topics       950 
12.2 Algebraic Techniques        953 
12.3 Power Series Methods      959 
12.4 Ordinary Points        965 
12.5 Regular Singular Points       968 
12.6 Bessel Functions       981 
12.7 Legendre Polynomials      985 
12.8 Orthogonality         996 

A Background Topics     1005   
A.1 Calculus        1005
A.2 Graphics        1015
A.3 Explicit and Implicit Answers       1024
A.4 Numerical and Graphical Answers     1029
A.5 Implicit Functions       1041