Differential Equations with Boundary Value Problems

1. Introduction to Differential Equations.


Differential Equation Models. The Derivative. Integration.



2. First-Order Equations.


Differential Equations and Solutions. Solutions to Separable Equations. Models of Motion. Linear Equations. Mixing Problems. Exact Differential Equations. Existence and Uniqueness of Solutions. Dependence of Solutions on Initial Conditions. Autonomous Equations and Stability. Project—The Daredevil Skydiver.



3. Modeling and Applications.


Modeling Population Growth. Models and the Real World. Personal Finance. Electrical Circuits. Project—The Spruce Budworm. Project—Social Security, Now or Later.



4. Second-Order Equations.


Definitions and Examples. Second-Order Equations and Systems. Linear, Homogeneous Equations with Constant Coefficients. Harmonic Motion. Inhomogeneous Equations; the Method of Undetermined Coefficients. Variation of Parameters. Forced Harmonic Motion. Project—Nonlinear Harmonic Oscillators.



5. The Laplace Transform.


The Definition of the Laplace Transform. Basic Properties of the Laplace Transform. The Inverse Laplace Transform. Using the Laplace Transform to Solve Differential Equations. Discontinuous Forcing Terms. The Delta Function. Convolutions. Summary. Project—Forced Harmonic Oscillators.



6. Numerical Methods.


Euler's Method. Runge-Kutta Methods. Numerical Error Comparisons. Other Solvers. A Cautionary Tale. Project—Numerical Error Comparison.



7. Matrix Algebra.


Vectors and Matrices. The Geometry of Systems of Linear Equations. Solving Systems of Equations. Properties of Solution Sets. Subspaces. Determinants.



8. An Introduction to Systems.


Definition and Examples. Geometric Interpretation of Solutions. Qualitative Analysis. Linear Systems. Project—Long Term Behavior of Solutions.



9. Linear Systems with Constant Coefficients.


Overview of the Technique. Planar Systems. Phase Plane Portraits. Higher-Dimensional Systems. The Exponential of a Matrix. Qualitative Analysis of Linear Systems. Higher-Order Linear Equations. Inhomogeneous Linear Systems. Project—Phase Plane Portraits. Project—Oscillations of Linear Molecules.



10. Nonlinear Systems.


The Linearization of a Nonlinear System. Long-Term Behavior of Solutions. Invariant Sets and the Use of Nullclines. Long-Term Behavior of Solutions to Planar Systems. Conserved Quantities. Nonlinear Mechanics. The Method of Lyapunov. Predator-Prey Systems. Project—Human Immune Response to Infectious Disease. Project—Analysis of Competing Species.



11. Power Series Solutions.


Review of Power Series. Series Solutions Near Ordinary Points. Legendre's Equation. Types of Singular Points—Euler's Equation. Series Solutions Near Regular Singular Points. Solutions in the Exceptional Cases. Bessel's Equation and Bessel Functions.



12. Fourier Series Methods.


Computation of Fourier Series. Convergence of Fourier Series. Fourier Cosine and Sine Series. The Complex Form of a Fourier Series. The Discrete Fourier Transform and the FFT.



13. Partial Differential Equations.


Derivation of the Heat Equation. Separation of Variables for the Heat Equation. The Wave Equation. Laplace's Equation. Laplace's Equation on a Disk. Sturm Liouville Problems. Orthogonality and Generalized Fourier Series. Temperature in a Ball—Legendre Polynomials. Time Dependent PDEs in Higher Dimension. Domains with Circular Symmetry—Bessel Functions.