Differential Equations

Definitions, sources and types of differential equation. Solutions. 1st order Ordinary Differential Equations (ODE). Integrals as solutions, slope fields, existence and uniqueness of the solution, Separable equations, implicit solutions,Integration factor, Substitutions, Bernoulli equations, homogeneous equations, Autonomous equations, Higher order linear ODEs Equations with constant coefficients, complex solutions and Euler, linear independence, Mechanicalvibrations, free oscillations, Non-homogeneous equations, Forced oscillations and resonance, Systems of ODEs, Eigenvalue methods, Two dimensional systems and vector fields, Second order equations and applications, Multiple eigenvalues, Matrix exponentials, Non-homogeneous systems of equations Fourier series, Boundary value problems, Fourier series, Sine and cosine series, Partial Differential Equations, Head equation, Wave equation,Laplace equation, Laplace Transform, Laplace transform and its inverse, Solution of ODEs with Laplace transforms,

Objectives

The main objective of the course is that the students acquire the ability to use differential equations to model physical phenomena, analyze their properties, solve such equations and use the solutions to deeper understanding of these phenomena. In particular, students will be able to Model physical phenomena with ordinary differential equations, partial differential equations and systems of differential equations , Solve such as the above mentioned equations using analytic or graphical methods , Analyze and communicate the results above mentioned results. Show clear understanding on Fourier Series analysis and Laplace transforms. Ability to use software systems like the http://www.wolframalpha.com/ and http://www.math.uiuc.edu/iode/ for all the above.

Prerequisites

Calculus I, Linear Algebra

Syllabus

Definitions, sources and types of differential equation. Solutions. 1st order Ordinary Differential Equations (ODE). Integrals as solutions, slope fields, existence and uniqueness of the solution, Separable equations, implicit solutions,Integration factor, Substitutions, Bernoulli equations, homogeneous equations, Autonomous equations, Higher order linear ODEs Equations with constant coefficients, complex solutions and Euler, linear independence, Mechanicalvibrations, free oscillations, Non-homogeneous equations, Forced oscillations and resonance, Systems of ODEs, Eigenvalue methods, Two dimensional systems and vector fields, Second order equations and applications, Multiple eigenvalues, Matrix exponentials, Non-homogeneous systems of equations Fourier series Boundary value problems, Fourier series, Sine and cosine series, Partial Differential Equations, Head equation, Wave equation,Laplace equation, Laplace Transform, Laplace transform and its inverse, Solution of ODEs with Laplace transforms,