# MA 221 - Differential Equations

MA 221 - Differential Equations

Ordinary differential equations of first and second order, homogeneous and non-homogeneous equations; improper integrals, Laplace transforms; review of infinite series, series solutions of ordinary differential equations near an ordinary point; boundary-value problems; orthogonal functions; Fourier series; separation of variables for partial differential equations.

**Prerequisites:** MA 116

Course Objectives

A first course in differential equations, MA221 introduces STEM students to techniques for solving linear scalar ODEs, special cases of nonlinear first-order scalar equations, and two-point boundary-value problems. Systems of ODEs are introduced in MA227. The course also provides a first introduction to PDEs with examples of linear equations solvable by separation of variables. MA221 is a core requirement in the science and engineering programs.

Learning Outcomes

Upon completing this course, it is expected that a student will be able to do the following:

**1. Mathematical Foundations**

**Classification of ODEs:**Classify scalar Ordinary Differential Equations (ODEs) as to linear vs. nonlinear, homogeneous vs. non-homogeneous, constant coefficient, separable, or exact.**First Order Scalar ODEs:**Derive solutions to first order scalar equations that are classified as linear, separable, or exact.**Second Order Linear ODEs:**Derive the general homogeneous solution to any constant coefficient, second order, linear ODE.**Non-homogeneuos Equations:**Solve non-homogeneous linear ODEs using the Method of Undetermined Coefficients and Variation of Parameters.**Laplace Transform:**Determine forward and inverse Laplace Transforms.**Power Series Methods:**Identify and classify singular points for linear ODEs and derive the power series solution in a neighborhood of an ordinary singular point.**Fourier Series:**Derive the Fourier coefficients for the sine and cosine series.

**2. Applications of Mathematics:**

**Application of Laplace Transform:**Apply Laplace Transform methods to solve initial value problems for constant coefficient linear ODEs.**Application of Fourier Series:**Apply Fourier Series methods to solve boundary value problems for linear ODEs.**Separation of Variables for PDEs:**Apply Separation of Variables to solve elementary examples of linear second order Partial Differential Equations (heat and wave equations).

Textbooks

Nagle, R. Kent, Edward B. Saff, and Arthur David Snider, *Fundamentals of Differential Equations and Boundary Value Problems*, Fifth edition, Addison Wesley.