In this lesson, we are introduced to the Laplace Transform and learn some basic formulas for its use. The Laplace Transform is the continuous analog of summation of a power series. By replacing N with a continuous variable T and using integration instead of summation, we can turn a function defined for positive values of T into a function of S. The goal is to make the integral converge, so we need X to be a positive number less than 1. By changing the variables and naming conventions, we end up with a function F of S, which is no longer a function of X. This is called a little plus transform.
Introduction to the Laplace Transform; Basic Formulas -- Lecture 19b. Learn about the Laplace Transform and some basic formulas for its use.
Arthur Mattuck, 18.03 Differential Equations, Spring 2006. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed November 27, 2008). License: Creative Commons BY-NC-SA.
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What is the most popular way of solving differential equations?
How can you derive the Laplace Transform?
How does the Laplace Transform relate to power series?
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This is a continuation of the previous lesson on the Laplace Transform derivation. We finally get the Laplace Transform at the end of this lecture. A must-see for those of you who want to know how to come up with this formula.