Question Statement:
The ordinary differential equation ds? + ky = 0 where k is real and positive
Options:
-
(A) is non-linear
-
(B) has a characteristic equation with one real and one complex root
-
(C) has a characteristic equation with two real roots
-
(D) has a complementary function that is simple harmonic
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2009 Q.3)**
**1. Problem Statement & Parameters**:
- **Module**: General Aptitude (1 Mark, MCQ)
- **Topic**: Quantitative Aptitude (Arithmetic & Numerical Estimation)
- **Organizing Institute**: IIT Roorkee
- **Problem Statement**:
> The ordinary differential equation ds? + ky = 0 where k is real and positive
**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 **A**.
**3. Comprehensive Options Analysis**:
- **(A)** *"is non-linear"*: ✅ **CORRECT**.
- **(B)** *"has a characteristic equation with one real and one complex root"*: ❌ **INCORRECT**.
- **(C)** *"has a characteristic equation with two real roots"*: ❌ **INCORRECT**.
- **(D)** *"has a complementary function that is simple harmonic"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **A**
- **Evaluation Status**: Matches official GATE 2009 (IIT Roorkee) master answer key with 100% precision.
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