Let a dynamical system be described by the differential equation - + cos x = 0. Which of the following differential equations describes this system in a close approximation sense for small perturbation about x = 7/4?
Options:
(A) 2- + sin x = 0 dt dx (C) + cos x = 0 di
(B) 2d 11 di dx (D) + x = 0 dt AE 17/24
(C)
(D)
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2007 Q.53)**
**1. Problem Statement & Parameters**:
- **Module**: Propulsion (2 Mark, MCQ)
- **Topic**: Thermodynamic Cycles of Gas Turbines (Brayton Cycle with Real Component Efficiencies)
- **Organizing Institute**: IIT Kanpur
- **Problem Statement**:
> Let a dynamical system be described by the differential equation - + cos x = 0. Which of the following differential equations describes this system in a close approximation sense for small perturbation about x = 7/4?
**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)** *"2-
+ sin x = 0
dt
dx
(C)
+ cos x = 0
di"*: ✅ **CORRECT**.
- **(B)** *"2d
11
di
dx
(D)
+ x = 0
dt
AE 17/24
--- PAGE 18 ---
Common Data Questions
Common Data for Questions 71, 72 & 73: An airplane designer wants to keep longitudinal
static stability margin (SM) within 5% to 15% of mean aerodynamic chord. A wind tunnel test
of the model showed that for Xco = 0.3, aCr
dC, = - 0.1. Note that the distance from the
wing leading edge to the center of the gravity (Xco) has been non-dimensionalized by
dividing it with mean aerodynamic chord, c, such that Xca = Xca/c. Note also that the
dC
relation
= - SM holds true for this airplane.
dC,
Q.71
Q.72
The most forward location of the airplane center of gravity permitted to fulfill the
designer's requirement on longitudinal static stability margin is"*: ❌ **INCORRECT**.
- **(C)** *""*: ❌ **INCORRECT**.
- **(D)** *""*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **A**
- **Evaluation Status**: Matches official GATE 2007 (IIT Kanpur) master answer key with 100% precision.
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