Dive into a comprehensive tutorial on Laplace Transforms, exploring their definition, properties, and applications in solving differential equations. Learn to compute Laplace transforms for various functions, including piecewise and trigonometric, and master inverse Laplace transforms using techniques like partial fraction decomposition and completing the square. Discover the linearity property, important transform pairs, and shifting theorems. Apply these concepts to solve initial value problems and work with unit step functions. Conclude with an in-depth proof of the convolution theorem, gaining a solid foundation in this powerful mathematical tool for engineering and physics applications.
Overview
Syllabus
The Definition of the Laplace Transform and Three Basic Examples.
Linearity Property of the Laplace Transform and 7 Useful Transforms to Know! Full Example..
Find the Laplace Transform of cos(kt) using the Definition.
Laplace Transform of a Piecewise Function Using the Definition.
Inverse Laplace Transform and Linearity of Inverse Laplace :: With Examples.
Use Laplace Transform to Solve Initial Value Problem :: Full Example with Partial Fraction Decomp.
Inverse Laplace Transform :: Completing the Square :: First Translation Theorem in Reverse.
Inverse Laplace Partial Fraction Decomposition :: Overall Strategy.
Shifting Laplace Transforms :: The First Translation Theorem for Laplace Transforms.
Laplace Transform With Unit Step Functions :: Second Shifting/Translation Theorem.
Proof of the Convolution Theorem :: Laplace Transforms.
Taught by
Jonathan Walters
