Lecture Notes: Laplace Transform Series - Introduction and Basic Examples

Overview

• Introduction to a new series on the Laplace transform.
• Focus on the usefulness of Laplace transform in solving differential equations.
• Objective of the first video: Definition and computation of Laplace transform through examples.

Why Laplace Transform?

• Useful for solving ordinary differential equations (ODEs) with initial conditions.
• Converts differential equations into algebraic equations (no derivatives involved).
• Simplifies solving process for algebraic equations, enabling conversion back to solutions for original differential equations.

Defining Laplace Transform

• Notation: Uses a squiggly L.
• Input: Function of time (t) ( f(t) ).
• Output: Function in terms of variable ( s ), denoted as ( F(s) ).
• Formula: [ F(s) = \int_0^\infty e^{-st} f(t) , dt ]
• Convergence: Depends on the values of ( s ) and ( a ) for different functions.

Examples

Example 1: Exponential Function

• Function: ( e^{at} ) where ( a ) is a constant.
• Laplace Transform: [ \int_0^\infty e^{-st} e^{at} , dt = \frac{1}{s-a} \text{ if } s > a ]
• Convergence: Depends on relation of ( s ) to ( a ).
  • Converges if ( s > a ).
  • Diverges if ( s \le a ).

Example 2: Step Function

• Function: Step function (Heaviside function) ( u(t-a) ).
• Behavior:
  • Zero for ( t < a ).
  • One for ( t \ge a ).
• Laplace Transform: [ F(s) = \frac{e^{-as}}{s} ]
• Application: Models functions with step discontinuities.

Example 3: Polynomial Function

• Function: ( t^n ).
• Use of Gamma Function: Defined as: [ \Gamma(x) = \int_0^\infty e^{-t} t^{x-1} , dt ]
  • ( \Gamma(1) = 1 )
  • ( \Gamma(n+1) = n! )
• Laplace Transform: [ F(s) = \frac{n!}{s^{n+1}} ]

Conclusion

• Overview of computing the Laplace transform through these examples.
• Preview of next video covering three properties of the Laplace transform.
• Encouragement to leave questions in the comments and explore linked courses.