Course Introduction

• Prerequisite：calculus
• Recommended for： sophomore
• Introduction：

Differential Equation is an immediate-connected course of Calculus and Linear Algebra, as its content consists of various applications of these basic mathematics. Differential Equation is originated from Newtonian mechanics and celestial mechanics, and still plays an important role in modern physics including quantum mechanics. On the other hand, mathematical modeling has prevailed in neural science, infectious diseases, gene networks, circuit theory, financial engineering, and traffic flows, etc. Although nonlinear differential equations can not be solved into solutions of exact forms in general, it is still feasible to apply various mathematical ideas and concepts such as analysis and geometry in understanding at least a certain part of the dynamics. Moreover, combining numerical computation with geometric delineation of or phase portraits enhances the investigation of differential equations.

• Syllabus：

1. First order ODEs
   existence and uniqueness theorem, separable equations, exact equations, numerical approximations
2. Second order linear ODES
   real and complex characteristic roots, integral transform, and oscillation
3. Higher order linear ODEs
   Wronskian, undetermined coefficient method, variation of parameters
4. Systems of linear ODEs
   fundamental matrix solutions and the exponential matrix solutions
5. Laplace transform
   basic Laplace transform and convolution integral
6. Nonlinear systems, stability, and asymptotic behaviors
7. Series solutions (optional)

• Reference：

1. Notes on Differential Equations by Jirí Lebl
2. Elementary Differential Equations and boundary value theorem by W. Boyce and R. DiPrima
3. Differential Equation by Blanchard, Devaney, Hall, and Lee.