📘 Laplace Transform – Additional Worked Problems (Extension Practice)
This section contains additional problems on Direct Laplace Transform and Inverse Laplace Transform. These problems strengthen algebra and conceptual clarity for GATE preparation.
🔹 Part A – Direct Laplace Transform Problems
Problem 1 – Polynomial Function
Find Laplace Transform of:
f(t) = 4t³ + 2t
L{tⁿ} = n! / sⁿ⁺¹
L{4t³} = 24/s⁴ L{2t} = 2/s²
Final Answer:
F(s) = 24/s⁴ + 2/s²
Problem 2 – Exponential with Sine
Find Laplace Transform of:
f(t) = e^{-3t} sin(4t)
L{e^{-at} sin(bt)} = b / [(s+a)² + b²]
Final Answer:
F(s) = 4 / [(s+3)² + 16]
Problem 3 – Differentiation Property
If F(s) = 5/(s+2), find Laplace of derivative.
L{df/dt} = sF(s) − f(0)
Assume initial value = 0
Final Answer:
5s/(s+2)
Problem 4 – Second Derivative
Find L{d²/dt² (t²)}
d²(t²)/dt² = 2 L{2} = 2/s
Final Answer:
2/s
🔹 Part B – Inverse Laplace Transform Problems
Problem 5 – Distinct Real Roots
Find inverse Laplace:
F(s) = (2s + 7)/[(s+1)(s+3)]
Assume A/(s+1) + B/(s+3)
Let s = -1 → A = 2 Let s = -3 → B = 1
Final Answer:
f(t) = 2e^{-t} + e^{-3t}
Problem 6 – Repeated Root
Find inverse Laplace:
F(s) = 6/(s+2)³
L⁻¹{1/(s+a)³} = t²/2 e^{-at}
Final Answer:
f(t) = 3t² e^{-2t}
Problem 7 – Quadratic Factor
Find inverse Laplace:
F(s) = (3s+6)/(s²+6s+13)
s²+6s+13 = (s+3)² + 2² Rewrite numerator = 3(s+3)
Final Answer:
f(t) = 3e^{-3t} cos(2t)
Problem 8 – Combined Factor
Find inverse Laplace:
F(s) = 4/[s(s+2)]
Assume A/s + B/(s+2)
A = 2 B = -2
Final Answer:
f(t) = 2 − 2e^{-2t}
Problem 9 – Initial & Final Value Theorem
Given:
F(s) = 8/(s(s+4))
Initial Value:
f(0⁺) = lim s→∞ [sF(s)] = 8
Final Value:
f(∞) = lim s→0 [sF(s)] = 2
🎯 Final Note
Strong command over partial fraction expansion and inverse Laplace is essential for solving transient analysis and control system problems. Practice repeatedly for speed and accuracy.
Laplace Mastery = Control + Network Mastery
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