### **Detailed Step-by-Step Solution (GATE AE 2025 Q.14)**
**1. Intuitive Physical Setup & Given Parameters**:
We are given the second-order partial differential equation:
$\frac{\partial^2 u}{\partial x^2} + 4\frac{\partial^2 u}{\partial x \partial y} + 2\frac{\partial^2 u}{\partial y^2} = 0$
and asked to classify its mathematical type.
**2. Step-by-Step Mathematical Derivation**:
1. **General 2nd-Order Linear PDE Canonical Form**:
$A \frac{\partial^2 u}{\partial x^2} + B \frac{\partial^2 u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2} + D\frac{\partial u}{\partial x} + E\frac{\partial u}{\partial y} + Fu = G$
2. **Extract Coefficients**:
Comparing with the given equation:
$A = 1, \quad B = 4, \quad C = 2$
3. **Compute Discriminant**:
$\Delta = B^2 - 4AC = (4)^2 - 4(1)(2) = 16 - 8 = 8$
4. **Classification Criteria**:
- If $B^2 - 4AC > 0 \implies$ **Hyperbolic** (e.g. wave equation, supersonic flow)
- If $B^2 - 4AC = 0 \implies$ **Parabolic** (e.g. heat equation, boundary layer equations)
- If $B^2 - 4AC < 0 \implies$ **Elliptic** (e.g. Laplace equation, subsonic potential flow)
Since $\Delta = 8 > 0$, the equation is **hyperbolic**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) elliptic**: ❌ **INCORRECT**. Requires $B^2 - 4AC < 0$.
- **(B) hyperbolic**: ✅ **CORRECT**. The discriminant $B^2 - 4AC = 16 - 8 = 8 > 0$, indicating hyperbolic character.
- **(C) parabolic**: ❌ **INCORRECT**. Requires $B^2 - 4AC = 0$.
- **(D) of mixed type**: ❌ **INCORRECT**. Coefficients are constants; therefore the sign does not change across the domain.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **B**
- **Exam Takeaway**: Always remember the discriminant sign: $B^2 - 4AC > 0$ is Hyperbolic (two real characteristic curves, wave-like);
lt; 0$ is Elliptic (no real characteristics, smooth boundary value problem); $= 0$ is Parabolic (one characteristic, diffusive).