Question

\(u (x,y)\) is governed by the following equation \[ \frac{\partial^{2}u}{\partial x^{2}} - 4\frac{\partial^{2}u}{\partial x \partial y} + 6\frac{\partial^{2}u}{\partial y^{2}} = x + 2y \] The nature of this equation is: 
 

• A. linear
• B. elliptic
• C. hyperbolic
• D. parabolic

Hint:

For PDEs: Elliptic if \(B^{2} - AC < 0\), Parabolic if \(B^{2} - AC = 0\), Hyperbolic if \(B^{2} - AC > 0\).

Answer 1

The Correct Option is B

Step 1: Identify coefficients. 
The PDE is of the form \[ A u_{xx} + 2B u_{xy} + C u_{yy}. \] Here, \(A = 1\), \(2B = -4 \Rightarrow B = -2\), \(C = 6\). 
 

Step 2: Use discriminant \(B^{2} - AC\). 
\[ B^{2} - AC = (-2)^{2} - (1)(6) = 4 - 6 = -2 < 0. \] Since the discriminant is negative, the equation is **elliptic**.