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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
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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.
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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.
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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)
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6. Differentiation and Integration Properties
L{f'(t)} = sF(s) - f(0) L{f''(t)} = s²F(s) - s f(0) - f'(0) Integration property: L{∫f(t) dt} = F(s)/s These are heavily used in solving differential equations.
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7. Unit Step Function
The unit step function is defined as: u(t - a) = 0 for t < a 1 for t ≥ a It helps represent piecewise-defined functions. Example: f(t) = u(t - 3) Means function starts at t = 3.
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8. Inverse Laplace Transform
Inverse Laplace converts F(s) back to f(t). L ¹{1/s} = 1⁻
L ¹{1/s²} = t⁻ L ¹{n!/sⁿ ¹} = tⁿ⁻ ⁺ L ¹{1/(s - a)} = e^at⁻ L ¹{a/(s² + a²)} = sin at⁻ L ¹{s/(s² + a²)} =cos at⁻
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9. Solved Examples
Example 1
Find L{5t³} L{t³} = 6/s⁴ Therefore: L{5t³} = 30/s⁴
Example 2
Find L{e²t + 3t} = 1/(s - 2) + 3/s²
Example 3
Find L{cos 4t} = s/(s² + 16)
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10. Solved Differential Equation
Solve: y' + 3y = 0 y(0) = 2
A) 1/s B) 1/s² C) s² D) 0 3.L{t²} equals A) 2/s³ B) 1/s³ C) 6/s³ D) s³ 4.L{sin at} equals A) a/(s² + a²) B) s/(s² + a²) C) 1/s D) s² 5.L{cos at} equals A) a/(s² + a²) B) s/(s² + a²) C) 1/s D) a/s 6.L{e^at} equals A) 1/(s + a) B) 1/(s - a) C) s - a D) s + a 7.L{t³} equals A) 6/s⁴ B) 3/s⁴ C) 1/s⁴ D) s⁴ 8.Inverse of 1/s² equals A) 1 B) t C) t² D) e^t 9.L{f'(t)} equals
A) F(s) B) sF(s) - f(0) C) F(s)/s D) s²F(s) 10.Unit step function equals A) Always 0 B) Always 1 C) 0 before a, 1 after D) Undefined 11.L{tⁿ} equals A) n!/sⁿ ¹⁺ B) n/s C) sⁿ D) 1/sⁿ 12.L{cosh at} equals A) s/(s² - a²) B) a/(s² - a²) C) s/(s² + a²) D) 1/s 13.L{sinh at} equals A) a/(s² - a²) B) s/(s² - a²) C) 1/s D) s² 14.Integration property gives A) F(s)/s B) sF(s) C) s²F(s) D) F(s) 15.Laplace is mainly used for A) Geometry B) Differential equations C) Statistics D) Probability 16.L{e³t} equals
5 B
6 B
7 A
8 B
9 B
10 C
11 A
12 A
13 A
14 A
15 B
16 B
17 A
18 B
19 A
20 A
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Final Revision Formula Sheet
L{1} = 1/s 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) Differentiation: sF(s) - f(0) Integration: F(s)/s
Revise formulas daily for scoring full marks in Laplace Transform
problems.
