Engineering Mathematics 2: Laplace Transform Revision

Welcome

• Lecturer: Gulshan Kumar
• Topic: Engineering Mathematics 2 - Laplace Transform (Unit 2)
• Purpose: One-shot revision of Laplace Transform focusing on important topics

Important Topics in Unit 2:

1. Application of Laplace Transform to Solve Ordinary Differential Equations (ODE)
2. Simultaneous Differential Equations using Laplace Transform
3. Convolution Theorem
4. Inverse Laplace Transform by Partial Fractions
5. Inverse Laplace Transform by Basic Formulas
6. Properties of Laplace and Inverse Laplace Transforms
7. Multiplication by T^n and Division by T
8. Unit Step Function (Heaviside function)
9. Periodic Functions

Key Points:

• Most Important Topics: Focus on differential equations and convolution theorem.
• Foundation: Master the starting concepts to understand advanced topics.
• Exam Questions: Emphasis on questions from final topics; they carry significant marks (±7 marks per question).

Basic Concepts of Laplace Transform:

1. Definition and Representation:

   • Converts a function from t-domain to s-domain.
   • Two representations: F(s) and f(s).
   • Integral form: (\mathcal{L}{f(t)} = \int_{0}^{\infty} e^{-st} f(t) , dt)
   • Important Formula: Integral Transform changes the function.

2. Initial and Final Value Theorems:

   • Initial Value Theorem: (\lim_{t\to 0} f(t) = \lim_{s\to \infty} sF(s))
   • Final Value Theorem: (\lim_{t\to \infty} f(t) = \lim_{s\to 0} sF(s))

3. Basic Formulas for Laplace Transform:

   • (\mathcal{L}{1} = \frac{1}{s})
   • (\mathcal{L}{t^n} = \frac{n!}{s^{n+1}}) for whole number n.
   • (\mathcal{L}{e^{at}} = \frac{1}{s-a})
   • Trigonometric Functions transformations.

4. Properties:

   • Linearity: (\mathcal{L}{af(t) + bg(t)} = a\mathcal{L}{f(t)} + b\mathcal{L}{g(t)})
   • First Shifting Theorem: (\mathcal{L}{e^{at}f(t)} = F(s-a))
   • Time Differentiation: (\mathcal{L}{f'(t)} = sF(s) - f(0))
   • Convolution: (\mathcal{L}{f(t) * g(t)} = F(s)G(s))
   • Unit Step Function: Definition and transformation.

Important Laplace Transform Applications:

1. Solving Ordinary Differential Equations (ODEs):

   • Use the transform to convert ODEs into algebraic equations.
   • Solve using manipulation and inverse Laplace.

2. Steps in Solving ODEs Using Laplace Transform:

   1. Take Laplace transform of both sides of ODE.
   2. Apply initial conditions and simplify.
   3. Solve for the Laplace transform of the solution.
   4. Invert the transform to get the solution.

3. Inverse Laplace Transform:

   • Identify and use basic inverse transformations.
   • Partial fractions decomposition for complex rational functions.
   • Common inverse transforms to memorize.

4. Convolution Theorem:

   • Use for complex inverse Laplace transforms.
   • Formula: (f(t) * g(t) = \int_{0}^{t} f(\tau)g(t-\tau) , d\tau)
   • Apply for functions that are products of transforms.
   • Regular practice of convolution problems increases understanding.

Additional Tips:

• Regular practice and revision are crucial to mastering Laplace Transforms.
• Ensure familiarity with all formulas and properties as they are extensively used in solving complex problems.
• Understand the derivation and application of each property to quickly identify their use in solving questions.

By following these organized notes and understanding the key points, you should be well-prepared for tackling Laplace Transform problems in your exams. Good luck!