The main reason for solving many differential equations is to try and learn something about an underlying physical process that the equation is believed to model. It is basic to the importance of differential equations that even the simplest equations correspond to useful physical models, for example, exponential growth and decay, spring-mass systems, or electrical circuits. Gaining an understanding of a complex natural process is usually accomplished by combining or building upon simpler and more basic models. Thus, a thorough knowledge of these models, the equations that describe them, and their solutions is the first and indispensable step toward the solution of more complex and realistic problems.
The successful student will firstly be capable of computing the solutions of various ordinary and partial differential equations that are useful in modeling natural phenomena. This proficiency is essential toward the greater aim, which is to help the student to understand something about mathematics and its role in science and modeling.
Students with prior knowledge of calculus who want to advance their skills and apply this to solution of real-world problems.
Boyce and DiPrima's "Elementary Differential Equations and Boundary Value Problems" Chapters 1-5 and 10
First order differential equations
Second order linear equations
Higher order linear equations
Series solutions of second order linear equations
At least three semesters of university calculus.