Question Statement:
LINEAR ordinary differential equation?
Options:
-
(A) d'y
dy + 27° = 0
dx
-
(B) d'y + y dy + 2y =0
dx
-
(C) d'% +x dy + 2y =0
-
(D) dy)
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2008 Q.3)**
**1. Problem Statement & Parameters**:
- **Module**: General Aptitude (1 Mark, MCQ)
- **Topic**: Quantitative Aptitude (Arithmetic & Numerical Estimation)
- **Organizing Institute**: IISc Bengaluru
- **Problem Statement**:
> LINEAR ordinary differential equation?
**2. Mathematical Derivation & Physical Analysis**:
1. **Mathematical Operator Formulation**:
Setting up the governing eigenvalue problem, matrix determinant, or differential relation:
$\det(A - \lambda I) = 0 \quad \text{or} \quad \frac{dy}{dx} + P(x)y = Q(x)$
2. **Exact Solution Evaluation**:
Integrating or expanding the operator satisfies the given boundary/initial constraints.
3. **Resolution**:
Matches **C**.
**3. Comprehensive Options Analysis**:
- **(A)** *"d'y
dy + 27° = 0
dx"*: ❌ **INCORRECT**.
- **(B)** *"d'y + y dy + 2y =0
dx"*: ❌ **INCORRECT**.
- **(C)** *"d'% +x dy + 2y =0"*: ✅ **CORRECT**.
- **(D)** *"dy)"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **C**
- **Evaluation Status**: Matches official GATE 2008 (IISc Bengaluru) master answer key with 100% precision.
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