\(u (x,y)\) 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:
\(u (x,y)\) 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:
Show Hint
- linear
- elliptic
- hyperbolic
- parabolic
The Correct Option is B
Solution and Explanation
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**.
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