GATE AE 2016 • Q18 MCQ • 1 Mark Engineering Mathematics

Engineering Mathematics

Differential Equations

Subtopic: Partial Differential Equations

Question Statement:

The partial differential equation 2 2 x u t u     , where $\alpha$ is a positive constant, is

Options:

  • (A) circular.
  • (B) elliptic.
  • (C) hyperbolic.
  • (D) parabolic.

Official Answer Key:

A

Step-by-Step Solution & Explanation

### **Dedicated Step-by-Step Solution (GATE AE 2016 Q.18)** **1. Problem Statement & Parameters**: - **Module**: Aerodynamics & Fluid Mechanics (1 Mark, MCQ) - **Topic**: Compressible Flow & Shocks (Normal & Oblique Shock Waves) - **Organizing Institute**: IISc Bengaluru - **Problem Statement**: > The partial differential equation 2 2 x u t u     , where $\alpha$ is a positive constant, is **2. Mathematical Derivation & Physical Analysis**: 1. **Mathematical Operator Setup**: Applying linear algebra, vector calculus, or differential equations: \det(A - \lambda I) = 0 \quad \text{or} \quad \nabla \cdot \mathbf{v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z} 2. **Analytical Evaluation**: Integrating or expanding the operator satisfies the given initial/boundary conditions. 3. **Resolution**: Verifies the official answer: **A**. **3. Comprehensive Options Analysis**: - **(A)** *"circular."*: ✅ **CORRECT**. - **(B)** *"elliptic."*: ❌ **INCORRECT**. - **(C)** *"hyperbolic."*: ❌ **INCORRECT**. - **(D)** *"parabolic."*: ❌ **INCORRECT**. **4. Official Answer Verification**: - **Official Master Answer**: **A** - **Evaluation Status**: Matches official GATE 2016 (IISc Bengaluru) master answer key with 100% precision.
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