Laplace Transforms and Solving ODEs

Explore the fundamental concepts and applications of Laplace Transforms in solving differential equations through this comprehensive video series. Delve into the extension of differential equations, focusing on the big ideas behind Laplace Transforms and their utility in solving equations with discontinuities or impulses. Learn about the properties of Laplace Transforms, including linearity, existence, and inverses, as well as their application to derivatives and integrals. Master the technique of using Laplace Transforms to solve differential equations through full examples and practice with translation and inverse transforms. Discover the convolution of functions, handle piecewise and discontinuous functions, and understand periodic functions in the context of Laplace Transforms. Gain insight into modeling impulses or infinite spikes using the Dirac Delta 'function' and apply this knowledge to solve ODEs with Delta functions using Laplace Transforms.

Intro to the Laplace Transform & Three Examples.
3 Properties of Laplace Transforms: Linearity, Existence, and Inverses.
The Laplace Transform of Derivatives and Integrals.
Using Laplace Transforms to solve Differential Equations ***full example***.
Translation and Inverse Laplace Transforms.
Inverse Laplace Transform Example using Partial Fractions.
How to use Laplace Transform to solve Differential Equations ** Full Example **.
The Convolution of Two Functions | Definition & Properties.
Laplace Transform and Piecewise or Discontinuous Functions.
Periodic Functions and the Laplace Transform.
The Dirac Delta 'Function': How to model an impulse or infinite spike.
Solving ODEs with Delta functions using Laplace Transforms.