Project ForgeEngineering Mathematics
Calculus
GATE 2023
Hard
Which one of the options given is the inverse Laplace transform of 1/(s³−s)? u(t) denotes the unit-step function.
- (−1 + (1/2)e⁻ᵗ + (1/2)eᵗ) u(t)
- B((1/3)e⁻ᵗ − eᵗ) u(t)
- C(−1 + (1/2)e^(−(t−1)) + (1/2)e^(t−1)) u(t−1)
- D(−1 − (1/2)e^(−(t−1)) − (1/2)e^(t−1)) u(t−1)
Explanation
1/(s³−s)=1/[s(s−1)(s+1)]. Partial fractions: A/s+B/(s−1)+C/(s+1) with A=−1, B=1/2, C=1/2 (evaluating the residues at s=0,1,−1). Taking the inverse transform of each term gives (−1+(1/2)eᵗ+(1/2)e⁻ᵗ)u(t), matching option (A).
Source: GATE ME 2023 Official Paper (2023)
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