Finding Particular Solutions to Inhomogeneous ODEs: Operator and Solution Formulas Involving Exponentials
In this lesson, we learn how to find particular solutions to second order equations with constant coefficients when the right-hand side is not zero. We focus on four types of inputs: simple exponential, decaying exponential, pure oscillation, and decaying oscillation. We use the exponential input theorem, which tells us that the particular solution for an exponential input is simply the exponential function divided by the polynomial evaluated at the exponent. We demonstrate how to apply this method to a specific example and find the general solution by taking the imaginary part of the complex solution.
Finding Particular Solutions to Inhomogeneous ODEs: Operator and Solution Formulas Involving Exponentials -- Lecture 13. A spattering of formulas for finding particular solutions to ODEs.
Arthur Mattuck, 18.03 Differential Equations, Spring 2006. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed November 26, 2008). License: Creative Commons BY-NC-SA.
More info at: http://ocw.mit.edu/terms
- Lecture Notes - Lecture Notes
- Muddy Card Responses - Muddy Card Responses are explanations of topics that were confusing to students
- Notes - Notes and exercises written by Prof. Mattuck
- Supplementary Notes - Chapter 10 Supplementary Notes written by Prof. Miller
- Recitation Problems - Recitation Problems and classwork exercises
- Recitation Solutions - Recitation problem solutions
- What is an inhomogeneous ODE?
- What is a particular solutions to an ODE?
- What is the exponential input theorem?
- What is the exponential shift rule?
- How do you know if a is a single or a double root?
-
Staff Review
This is a very important lesson in exponential differential equations and oscillations of springs. Many solution formulas and their proofs are presented in the lesson as well. A bit of a complicated and very theoretical lecture with many formulas represented.
