1
Advanced Engineering Mathematics
Comprehensive Class Notes
Chapter 5: Laplace Transforms — Solving Differential Equations
Know-How | Student Edition
, 2
Chapter Overview
Laplace transforms convert differential equations into algebraic equations. This allows us to
solve problems involving forcing, discontinuities, and impulses.
Key ideas:
• • transform from time-domain to s-domain
• • solve algebraic equation
• • inverse transform to return to original variable
1. Definition
The Laplace transform of f(t) is defined by
∞
ℒ{𝑓(𝑓)} = ∫ 𝑒 −𝑠𝑡 . 𝑓(𝑡) 𝑑𝑡
0
The exponential term damps future values so the integral converges.
2. Basic Transform Rules
L{1} = 1/s
ℒ{𝑒 𝑎𝑡 } = 1/(𝑠 − 𝑎)
L{sin bt} = b/(s² + b²)
Know-How | Student Edition
Advanced Engineering Mathematics
Comprehensive Class Notes
Chapter 5: Laplace Transforms — Solving Differential Equations
Know-How | Student Edition
, 2
Chapter Overview
Laplace transforms convert differential equations into algebraic equations. This allows us to
solve problems involving forcing, discontinuities, and impulses.
Key ideas:
• • transform from time-domain to s-domain
• • solve algebraic equation
• • inverse transform to return to original variable
1. Definition
The Laplace transform of f(t) is defined by
∞
ℒ{𝑓(𝑓)} = ∫ 𝑒 −𝑠𝑡 . 𝑓(𝑡) 𝑑𝑡
0
The exponential term damps future values so the integral converges.
2. Basic Transform Rules
L{1} = 1/s
ℒ{𝑒 𝑎𝑡 } = 1/(𝑠 − 𝑎)
L{sin bt} = b/(s² + b²)
Know-How | Student Edition
