The GATE Grind

GATE 2024 CE (CE1) – Question 28

Engineering Mathematics · PDE: Fourier series, separation of variables, diffusion, wave and Laplace equations · 1 mark · Multiple select

For the following partial differential equation, $x\frac{\partial^2f}{\partial x^2} + y\frac{\partial^2f}{\partial y^2} = \frac{x^2 + y^2}{2}$, which of the following option(s) is/are CORRECT?

  1. elliptic for x > 0 and y > 0
  2. parabolic for x > 0 and y > 0
  3. elliptic for x = 0 and y > 0
  4. hyperbolic for x < 0 and y > 0

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Show answer and explanation

Correct answer: (A) elliptic for x > 0 and y > 0; (D) hyperbolic for x < 0 and y > 0

Explanation

With $A = x$, $B = 0$ and $C = y$, the discriminant is $B^2 - 4AC = -4xy$. For $x > 0$ and $y > 0$ it is negative, so the equation is elliptic (A true, B false). For $x = 0$ the discriminant is zero, so it is parabolic, not elliptic (C false). For $x < 0$ and $y > 0$, $xy < 0$ and the discriminant is positive, so it is hyperbolic (D true).