Teaching Haryana (HPSC) Assistant Professor Mock Test 2025 Mathematical Science Partial Differential Equations Classification of Second Order PDE
Consider the PDE
P(x, y)\(\frac{\partial^2 u}{\partial x^2}\) + \(e^{x^2} e^{y^2} \frac{\partial^2 u}{\partial x \partial y}\) + Q(x,y) \(\frac{\partial^2 u}{\partial y^2}\) + \(e^{2 x} \frac{\partial u}{\partial x}\) + \(e^y \frac{\partial u}{\partial y}\) = 0
where P and Q are polynomials in two variables with real coefficients. Then which of the following is true for all choices of P and Q?
1
There exists R > 0 such that the PDE is elliptic in {(x, y) ∈ ℝ2: x2 + y2 > R}
2
There exists R > 0 such that the PDE is hyperbolic in {(x, y) ∈ ℝ2 :x2 + y2 > R}
3
There exists R > 0 such that the PDE is parabolic in {(x, y) ∈ ℝ2: x2 + y2 > R}
4
There exists R > 0 such that the PDE is hyperbolic in {(x, y) ∈ ℝ2: x2 + y2 < R}
5
Question Not Attempted