📘 Inverse Laplace Transform & Partial Fraction Expansion (Deep Practice)
Inverse Laplace Transform is extremely important in GATE. Most mistakes happen in partial fraction expansion. This chapter gives deep numerical practice.
🔹 1. Basic Inverse Transform Pairs
L⁻¹{1/s} = 1 L⁻¹{1/s²} = t L⁻¹{1/(s+a)} = e^{-at} L⁻¹{a/(s²+a²)} = sin(at) L⁻¹{s/(s²+a²)} = cos(at)
🔹 2. Partial Fraction Method (Distinct Real Roots)
Example 1
Find inverse Laplace of:
F(s) = (3s + 5)/[(s+1)(s+2)]
Step 1: Assume
(3s+5)/[(s+1)(s+2)] = A/(s+1) + B/(s+2)
Step 2: Multiply both sides
3s+5 = A(s+2) + B(s+1)
Step 3: Solve coefficients
Let s = -1 → A = 2 Let s = -2 → B = 1
Step 4: Inverse Transform
f(t) = 2e^{-t} + e^{-2t}
🔹 3. Repeated Roots
Example 2
Find inverse Laplace:
F(s) = 5 / (s+3)²
We know:
L⁻¹{1/(s+a)²} = t e^{-at}
Therefore:
f(t) = 5t e^{-3t}
🔹 4. Quadratic Factor (Complex Roots)
Example 3
Find inverse Laplace:
F(s) = (2s+4)/(s²+4s+13)
Rewrite denominator:
s²+4s+13 = (s+2)² + 3²
Rewrite numerator:
2(s+2)
Thus:
F(s) = 2(s+2)/[(s+2)²+3²]
Inverse:
f(t) = 2e^{-2t} cos(3t)
🔹 5. Combined Repeated + Quadratic
Example 4
F(s) = (s+4)/[(s+1)(s²+4)]
Assume:
A/(s+1) + (Bs + C)/(s²+4)
Solve coefficients (algebra steps omitted here for brevity). Final result:
f(t) = Ae^{-t} + B cos(2t) + C sin(2t)
🔹 6. Initial Value Theorem Example
F(s) = 5/(s(s+2))
Initial value:
f(0⁺) = lim s→∞ [sF(s)] = lim s→∞ [5/(s+2)] = 5
🔹 7. Final Value Theorem Example
F(s) = 10/(s(s+5))
Final value:
f(∞) = lim s→0 [sF(s)] = lim s→0 [10/(s+5)] = 2
🔹 8. GATE Trap Problems
- Forgetting repeated root expansion
- Wrong numerator assumption for quadratic
- Sign errors in coefficient solving
- Applying Final Value Theorem when unstable
🎯 Final Summary
Strong partial fraction skills are essential. Most Laplace questions in GATE test algebra accuracy. Repeated practice improves solving speed significantly.
Master Partial Fractions → Master Laplace Transform
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