Module 6: Laplace Transforms
Master the Laplace transform: a powerful algebraic tool that converts differential equations into algebraic equations, making IVPs much easier to solve.
Your Progress
Lessons 1-4
Practice
Module Quiz
Review
Learning Objectives
By the end of this module, you will be able to:
- Compute Laplace transforms of basic functions using the definition and a table
- Find inverse Laplace transforms using partial fractions and completing the square
- Solve initial value problems by transforming both sides and inverting
- Handle discontinuous forcing functions using the Heaviside step function and second shifting theorem
- Apply the Laplace transform to convert differential equations into algebraic equations
Module Lessons
1
Definition and Basic Transforms
The Laplace integral, transforms of elementary functions, linearity, and building a transform table.
45-55 minutes
2
Inverse Laplace and Partial Fractions
Recovering f(t) from F(s) using partial fraction decomposition and completing the square.
45-55 minutes
3
Solving IVPs with Laplace Transforms
Transform both sides, solve for Y(s), invert to get y(t). Full solution procedure.
40-50 minutes
4
Step Functions and Discontinuous Forcing
Heaviside function u(t-a), second shifting theorem, piecewise forcing terms.
40-50 minutes