Ordinary differential equations serve as models for numerous real-world applications and have led to many significant insights into nature and technology. Examples are the spreading of infectious diseases, the immune response to bacterial and viral infections, transport of particles in the atmosphere and the ocean (eg to understand how ash from wild fires spreads and how to best respond to oil spills), prediction of rapid changes in climate and ecological models, HIV drug therapies, and the distribution of wealth in societies.
By the end of the course, you will be able to
formulate questions about real-world problems and create ordinary differential equations models to answer them
determine when an ordinary differential equation has a solution and when the solution is unique
analyse and solve ordinary differential equations using qualitative, analytical, and numerical techniques
draw conclusions about real-world problems from ordinary differential equations models
formulate and interpret mathematical statements
support arguments using the theoretical foundations of differential equations
We will cover the following content in APMA 0355 (with applications, programming, theory, and optional topics highlighted in different colors):
Definition of solutions
Classification of differential equations
Intuition into first-order ODEs
Euler's method and codes
Modeling: compartment models and population dynamics
Introduction to mathematical statements
Existence and uniqueness theorem
Qualitative theory + applications interweaved
Direction fields
Separation of variables
Linear scalar ODEs
Applications: mixing, mechanics, populations
Proof of uniqueness: Gronwall's lemma
Outline of existence proof
Linear algebra review
Linear systems: general solution, superposition
Diagonalization and matrix exponential
Repeated eigenvalues
Inhomogeneous systems: Variation of constants
Liouville's formula
Nonlinear planar ODEs: phase-plane analysis
Applications in climate, ecology, epidemiology
Second-order equations
Boundary-value problems
Introduction to PDEs and Fourier series