The two-dimensional linearized potential flow equation for steady subsonic flow is given by: $$\beta^2 \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0, \quad \text{where } \beta^2 = 1 - M_\infty^2 > 0$$ This partial differential equation is classified as:

(A) Elliptic PDE

(B) Hyperbolic PDE

(C) Parabolic PDE

(D) Nonlinear Burgers equation

Answer Key:

A

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: PDE Standard Form:** $A \phi_{xx} + B \phi_{xy} + C \phi_{yy} = 0$. 2. **Stage 2: Coefficients:** $A = 1 - M_\infty^2 > 0$, $B = 0$, $C = 1$. 3. **Stage 3: Discriminant:** $\Delta = B^2 - 4AC = -4(1 - M_\infty^2) < 0$. 4. **Stage 4: Classification:** $\Delta < 0$ corresponds strictly to an elliptic PDE. 5. **Stage 5: Selection:** Option A is correct.

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