GATE BT 2019 • Q49
MCQ • 2 Marks
Section 1: Engineering Mathematics
Differential Equations
Subtopic: Fundamentals of Differential Equations
Question Statement:
The Laplace transform of the polynomial function f(t) = t^2 + 2t + 1 for s > 0 is:
Options:
-
(A) (2/s^3) + (2/s^2) + (1/s)
-
(B) (1/s^3) + (2/s^2) + (1/s)
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(C) (2/s^2) + (2/s) + 1
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(D) (2/s^3) + (1/s^2) + (2/s)
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2019 Q.49)**
**1. Governing Concept**:
- **Topic**: Engineering Mathematics & Laplace Transforms
- Polynomial function: $f(t) = t^2 + 2t + 1$.
- Standard Laplace transform formulas for $s > 0$:
$\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}, \quad \mathcal{L}\{1\} = \frac{1}{s}$
- Applying linearity:
$\mathcal{L}\{t^2\} = \frac{2!}{s^3} = \frac{2}{s^3}$
$\mathcal{L}\{2t\} = 2 \cdot \frac{1!}{s^2} = \frac{2}{s^2}$
$\mathcal{L}\{1\} = \frac{1}{s}$
- Summing the terms:
$F(s) = \frac{2}{s^3} + \frac{2}{s^2} + \frac{1}{s} = \frac{2 + 2s + s^2}{s^3} = \frac{s^2 + 2s + 2}{s^3}$
**2. Option Analysis**:
- **Option (A) is INCORRECT**: Incorrect powers of $s$.
- **Option (B) is INCORRECT**: Numerator arithmetic error.
- **Option (C) is INCORRECT**: Missing factorial on $t^2$.
- **Option (D) [2/s³ + 2/s² + 1/s] is CORRECT**: Exact application of standard Laplace transform tables.
**3. Final Answer**:
- **Correct Choice**: **D**
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