MATH 151
Course Description
Calculus paved the way for the systematic development of the subject Differential Equations, which is now a well-established and well-studied advanced topic in mathematics. The study of this subject goes beyond the realm of mathematics. It is now a prominent practice to use differential equations in applied sciences as a modeling approach to yield mathematical explanations to a scientific phenomenon. For instance, the projections made about the spread of the recent pandemic were obtained using differential equation models.
In this course, we will study the fundamental type of differential equations known as ordinary differential equations (ODEs). These equations only involve full derivatives of unknown functions. You will gain a comprehensive knowledge about ODEs as we discuss its theoretical framework, solution methods, and modeling applications. This course makes you appreciate your learnings in previous Calculus courses as you work through the problem sets on analyzing and solving different classes of ODEs. You will also see the world through the lens of ODEs as we tackle real-life problems via systems modeling approach. Examples of applications of ODEs that will be discussed here are, but not limited to, population dynamics, chemical reactions, and mechanical vibrations. You will demonstrate your understanding about ODEs through the different learning tasks, exercises, and assignments.
Course Learning Outcomes
After completing this course, you should be able to analyze ODEs and apply them in
modelling real-life systems. Specifically, you should be able to:
1) describe the different types of differential equations and their nature of solutions;
2) solve various types of differential equations using graphical and theoretical approaches;
3) formulate mathematical models in terms of differential equations.
Course Outline
I. Introduction
A. Definition of ODEs and some terminologies
B. ODEs in real-life problems
C. Classification of ODEs
II. First-order ODEs: Solution approaches
A. Initial value problems (IVPs) and other examples
B. Direction/vector fields and integral curves
C. Phase lines
D. A numerical method: Euler’s approximation
III. First-order ODEs: Analytic methods
A. Separable equations
B. Exact equations
C. Linear ODEs
D. Special integrating factors
E. Substitutions and transformations
IV. Modelling first-order ODEs
A. Linear models
B. Nonlinear models
C. Systems of first-order ODEs
V. Homogeneous linear second-order ODEs with constant coefficients
A. Characteristic equations with real roots
B. Characteristic equations with complex roots
C. Mechanical vibrations: free oscillators
VI. Homogeneous linear second-order ODEs with nonconstant coefficients
A. Reduction of order
B. Fundamental sets of solutions
C. The Wronskian
VII. Nonhomogeneous linear second-order ODEs
A. Method of undetermined coefficients
B. Variation of parameters
C. Forced mechanical vibrations
D. Cauchy-Euler equations
VIII. Laplace transforms
A. Laplace transform and its inverse
B. Table of Laplace transforms
C. Solving IVPs using Laplace transforms
D. Dirac delta functions
E. Convolution integrals
IX. First-order linear systems of ODEs
A. Transforming higher-order ODEs to a system of ODEs
B. Solving systems of ODEs by elimination
C. Eigenvalue/eigenvector method for solving homogeneous linear systems with
constant coefficients
D. Qualitative solution behavior: Phase portraits
E. The matrix exponential function
F. Nonhomogeneous systems
G. Solving higher-order linear ODEs (optional)