First-Order Differential Equations / 1: |
Differential Equations and Mathematical Model |
Integrals as General and Particular Solutions |
Direction Fields and Solution Curves |
Separable Equations and Applications |
Linear First-Order Equations |
Substitution Methods and Exact Equations |
Mathematical Models and Numerical Methods / 2: |
Population Models |
Equilibrium Solutions and Stability |
Acceleration-Velocity Models |
Numerical Approximation: Euler's Method |
A Closer Look at the Euler Method |
The Runge-Kutta Method |
Linear Systems and Matrices / 3: |
Introduction to Linear Systems |
Matrices and Gaussian Elimination |
Reduced Row-Echelon Matrices |
Matrix Operations |
Inverses of Matrices |
Determinants |
Linear Equations and Curve Fitting |
Vector Spaces / 4: |
The Vector Space R |
The Vector Space Rn and Subspaces |
Linear Combinations and Independence of Vectors |
Bases and Dimension for Vector Spaces |
General Vector Spaces |
Linear Equations of Higher Order / 5: |
Introduction: Second-Order Linear Equations |
General Solutions of Linear Equations |
Homogeneous Equations with Constant Coefficients |
Mechanical Vibrations |
Undetermined Coefficients and Variation of Parameters |
Forced Oscillations and Resonance |
Eigenvalues and Eigenvectors / 6: |
Introduction to Eigenvalues |
Diagonalization of Matrices |
Applications Involving Powers of Matrices |
Linear Systems of Differential Equations / 7: |
First-Order Systems and Applications |
Matrices and Linear Systems |
The Eigenvalue Method for Linear Systems |
Second-Order Systems and Mechanical Applications |
Multiple Eigenvalue Solutions |
Numerical Methods for Systems |
Matrix Exponential Methods / 8: |
Matrix Exponentials and Linear Systems |
Nonhomogeneous Linear Systems |
Spectral Decomposition Methods |
Nonlinear Systems and Phenomena / 9: |
Stability and the Phase Plane |
Linear and Almost Linear Systems |
Ecological Models: Predators and Competitors |
Nonlinear Mechanical Systems |
Laplace Transform Methods / 10: |
Laplace Transforms and Inverse Transforms |
Transformation of Initial Value Problems |
Translation and Partial Fractions |
Derivatives, Integrals, and Products of Transforms |
Periodic and Piecewise Continuous Forcing Functions |
Power Series Methods / 11: |
Introduction and Review of Power Series |
Power Series Solutions |
Frobenius Series Solutions |
Bessel's Equation |
References for Further Study |
The Existence and Uniqueness of Solutions / Appendix A: |
Theory of Determinants / Appendix B: |
Answers to Selected Problems |
Index |
First-Order Differential Equations / 1: |
Differential Equations and Mathematical Model |
Integrals as General and Particular Solutions |