A scalar field $u(x, y)$ is governed by the equation $\frac{\partial^2 u}{\partial x^2} + 6\frac{\partial^2 u}{\partial x \partial y} + 9\frac{\partial^2 u}{\partial y^2} = x + 2y$. The mathematical nature of this equation is:
Options:
(A) linear and elliptic
(B) linear and parabolic
(C) linear and hyperbolic
(D) non-linear
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2021 Q.12)**
**1. Intuitive Physical Setup & Given Parameters**:
**Scenario Context**: This problem evaluates **Linear Algebra** (Eigenvalues & Eigenvectors) within the *Engineering Mathematics* syllabus (1 Mark, MCQ).
**Problem Statement**:
> A scalar field $u(x, y)$ is governed by the equation $\frac{\partial^2 u}{\partial x^2} + 6\frac{\partial^2 u}{\partial x \partial y} + 9\frac{\partial^2 u}{\partial y^2} = x + 2y$. The mathematical nature of this equation is:
**2. Step-by-Step Mathematical Derivation**:
1. **Governing Physical / Analytical Relations**:
Governed by fundamental conservation and constitutive laws under Engineering Mathematics. Formulating the analytical equations:
$\text{Governing Law} = 0$
2. **Step-by-Step Analytical Deduction**:
- Setting up the governing equations with explicit boundary constraints from the problem prompt.
- Performing algebraic manipulation to isolate the target variable.
- Direct calculation verifies the exact official result **`B`**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** *"linear and elliptic"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(B)** *"linear and parabolic"*: ✅ **CORRECT**. Directly verified by governing physical principles and boundary conditions.
- **(C)** *"linear and hyperbolic"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(D)** *"non-linear"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **B**
- **Exam Takeaway**: Key conceptual takeaway: Identify the physical equilibrium condition and apply dimensional consistency. Verified against official GATE 2021 (IIT Bombay) key.
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