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