Engineering Mathematics 2 - Laplace Transform

Introduction

• After Unit 1, Unit 2: Laplace Transform
• Here are three main sections: Detailed Topics, Revision, and Important Topics
• Unit 4, 5, and 1 have already had a one-shot revision.
• Focus: Revision of Unit 2 (Laplace Transform)

Unit 2 Topics:

1. Application of Laplace Transform to solve Ordinary Differential Equations
2. Convolution Theorem
3. Inverse Laplace Transform by Partial Fractions
4. Laplace Transform of Periodic Functions

Important Formulas and Properties

Basic Laplace Transform Formulas:

1. Transform of Simple Functions:
   • Laplace of 1 = 1/s
   • Laplace of T^n
2. Exponential Functions:
   • e^(at) -> 1/(s-a)
3. Trigonometrical Functions:
   • sin(at) -> a/(s² + a²)
   • cos(at) -> s/(s² + a²)
4. Transformation Rules:
   • Compound examples of sinh(at) and cosh(at)
5. Common Transform Examples

Important Properties:

1. Linearity Property:
   • L{af(t) + bg(t)} = aL{f(t)} + bL{g(t)}
2. First Shifting Property:
   • e^(at)f(t) -> F(s-a)
3. Second Shifting Property (Unit step function):
   • L{f(t-a)u(t-a)} = e^(-as)F(s)
4. Change of Scale Property:
   • F(s/a)/|a|

Key Topics and Their Revisions:

Application of Laplace Transform:

• Most Important Topics:
  • Solution of ordinary differential equations
  • Solution of simultaneous differential equations
  • Typical exam questions (7 marks)
• Key Points:
  • Starting topics necessary to understand advanced
  • Sequential revision:
    1. Basic formulas
    2. Essential properties
    3. Detailed derivations

Convolution Theorem:

• Used to find the inverse Laplace transform.
• Formula: [ (f * g)(t) = ∫_0^t f(τ)g(t-τ)dτ ]
• To solve inverse Laplace using convolution theorem, break the given equation:
  1. Identify F(s) and G(s) components
  2. Apply inverse Laplace
  3. Use convolution formula

Inverse Laplace Transform using Partial Fractions:

1. Linear Factors:
   • Example: 1/(s-1)(s-2)
2. Repeated Linear Factors:
   • Example: 1/(s-1)²
3. Quadratic Factors:
   • Example: 1/(s²+ k²)
4. Combination of Factors
   • Mixed form of the above
   • Method: Decompose into partial fractions, then solve each part

Laplace Transform of Periodic Functions:

• Useful for functions repeating over intervals.
• Formula and Steps to find: [ L{f(t)} = 1/(1 - e^{-Ts}) \times \text{integral from 0 to T} {e^{-st}f(t)dt} ]

Detailed Revisions: Examples and Derivations

1. Step-by-Step Derivations
2. Exam-style Questions
3. Convolution and Periodic Function Exercises

Practical Examples:

1. Solving differential equations
   • How to apply Laplace transforms to solve ordinary differential equations (ODEs)
   • Use initial conditions
2. Inverse Laplace transforms
   • Using properties and partial fractions
3. Convolution theorem applications
   • Practical examples of using convolution theorem to find inverse laplace
4. Periodic functions
   • Steps to calculate Laplace transforms for periodic functions

Conclusion

• Ensure understanding of each property and formula
• Consistent practice of problems – both initial and advanced levels
• Practice problems for each formula

Notes

• Regular revision sessions essential
• Solve and revisit complex problems
• Use properties effectively during problem-solving

Advance Preparation

• Detailed study of each unit
• Ensure coverage of all provided topics and subtopics

Regular practice and revision of these fundamentals ensure strong foundational knowledge essential for full comprehension and problem-solving capabilities in Laplace transforms.