Engineering Mathematics
Laplace Transform – Complete Concept + Solved Problems + MCQs
This study guide is designed for:
• Engineering semester exams
• Competitive technical exams
• Quick revision before tests
• Understanding differential equation solving
This PDF includes:
● Theory with explanation
● All important formulas
● Properties and theorems
● Solved numerical problems
● Differential equation solution
● 20 carefully selected MCQs
● Final revision formula sheet
PAGE 2
1. Introduction to Laplace Transform
Laplace Transform is a mathematical tool used to convert a function of time, f(t), into a function
of complex frequency, F(s).
It transforms differential equations into algebraic equations, which are easier to solve.
Mathematical Definition
L{f(t)} = ∫₀^∞ e^(-st) f(t) dt
Where:
t = time variable
s = complex number (s = σ + jω)
f(t) = time domain function
, F(s) = Laplace transform of f(t)
Laplace Transform exists if the integral converges.
PAGE 3
2. Conditions for Existence
The Laplace Transform of f(t) exists if:
● f(t) is piecewise continuous.
● f(t) is of exponential order.
● The integral converges for Re(s) > a.
Functions like polynomials, exponentials, sine and cosine generally satisfy these conditions.
PAGE 4
3. Standard Laplace Transforms
L{1} = 1/s
L{t} = 1/s²
L{t²} = 2/s³
L{t³} = 6/s⁴
L{tⁿ} = n!/sⁿ⁺¹
L{e^at} = 1/(s - a)
L{sin at} = a/(s² + a²)
L{cos at} = s/(s² + a²)
L{sinh at} = a/(s² - a²)
L{cosh at} = s/(s² - a²)
These formulas must be memorized for fast problem solving.
Laplace Transform – Complete Concept + Solved Problems + MCQs
This study guide is designed for:
• Engineering semester exams
• Competitive technical exams
• Quick revision before tests
• Understanding differential equation solving
This PDF includes:
● Theory with explanation
● All important formulas
● Properties and theorems
● Solved numerical problems
● Differential equation solution
● 20 carefully selected MCQs
● Final revision formula sheet
PAGE 2
1. Introduction to Laplace Transform
Laplace Transform is a mathematical tool used to convert a function of time, f(t), into a function
of complex frequency, F(s).
It transforms differential equations into algebraic equations, which are easier to solve.
Mathematical Definition
L{f(t)} = ∫₀^∞ e^(-st) f(t) dt
Where:
t = time variable
s = complex number (s = σ + jω)
f(t) = time domain function
, F(s) = Laplace transform of f(t)
Laplace Transform exists if the integral converges.
PAGE 3
2. Conditions for Existence
The Laplace Transform of f(t) exists if:
● f(t) is piecewise continuous.
● f(t) is of exponential order.
● The integral converges for Re(s) > a.
Functions like polynomials, exponentials, sine and cosine generally satisfy these conditions.
PAGE 4
3. Standard Laplace Transforms
L{1} = 1/s
L{t} = 1/s²
L{t²} = 2/s³
L{t³} = 6/s⁴
L{tⁿ} = n!/sⁿ⁺¹
L{e^at} = 1/(s - a)
L{sin at} = a/(s² + a²)
L{cos at} = s/(s² + a²)
L{sinh at} = a/(s² - a²)
L{cosh at} = s/(s² - a²)
These formulas must be memorized for fast problem solving.
