1. Introduction. Prologue: What Are Differential Equations? Four Introductory Models. Fundamental Concepts and Terminology.
2. Linear First-Order Equations. Methods of Solution. Some Elementary Applications. Generalized Solutions.
3. Nonlinear First-Order Equations I. Direction Fields and Numerical Approximation. Separable Equations. Bernoulli and Riccati Equations. Reduction of Order. Nonlinear First-order Equations in Applications.
4. Nonlinear First-Order Equations II. Construction of Local Solutions. Existence and Uniqueness. Qualitative and Asymptotic Behavior. The Logistic Population Model. Numerical Methods. A First Look at Systems.
5. Linear Second-Order Equations I. Introduction: Modeling Vibrations. State Variables and Numerical Approximation. Operators and Linearity. Solutions and Linear Independence. Variation of Constants and Green's Functions. Power-Series Solutions. Polynomial Solutions.
6. Linear Second-Order Equations II. Homogeneous Equations with Constant Coefficients. Exponential Shift. Complex Roots. Real Solutions from Complex Solutions. Unforced Vibrations. Periodic Force and Response.
7. The Laplace Transform. Definition and Basic Properties. More Transforms and Further Properties. Heaviside Functions and Piecewise-Defined Inputs. Periodic Inputs. Impulses and the Dirac Distribution. Convolution.
8. Linear First-Order Systems. Introduction. Two Ad Hoc Methods. Vector-Valued Functions and Linear Independence. Evolution Matrices and Variation of Constants. Autonomous Systems: Eigenvalues and Eigenvectors.
eAT and the Cayley-Hamilton Theorem. Asymptotic Stability.
9. Geometry of Autonomous Systems in the Plane. The Phase Plane. Phase Portraits of Homogeneous Linear Systems. Phase Portraits of Nonlinear Systems. Limit Cycles. Beyond the Plane.
10. Nonlinear Systems in Applications. Lotka-Volterra Systems in Ecology. Infectious Disease and Epidemics. Other Biological Models. Chemical Systems. Mechanics.
11. Diffusion Problems and Fourier Series. The Basic Diffusion Problem. Solutions by Separation of Variables. Fourier Series. Fourier Sine and Cosine Series. Sturm-Liouville Eigenvalue Problems. Singular Sturm-Liouville Problems. Eigenfunction Expansions.
12. Further Topics in PDEs. The Wave Equation. The 2-D Laplace Equation. The 2-D Diffusion Equation.
Appendices. 