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GATE 2021 AE – Question 12

Engineering Mathematics · Differential Equations: Classification of PDEs, wave, Laplace and heat equations by separation of variables · 1 mark · Multiple choice

The partial differential equation $u_{xx}-4u_{xy}+6u_{yy}=x+2y$ is

  1. linear
  2. elliptic
  3. hyperbolic
  4. parabolic

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Show answer and explanation

Correct answer: (B) elliptic

Explanation

For a second-order PDE written as $A\,u_{xx}+2B\,u_{xy}+C\,u_{yy}+\dots=0$, the type is decided by $B^2-AC$:
- $B^2-AC<0$: elliptic,
- $B^2-AC=0$: parabolic,
- $B^2-AC>0$: hyperbolic.

**Coefficients.** $u_{xx}-4u_{xy}+6u_{yy}=x+2y$ gives $A=1$, $2B=-4\Rightarrow B=-2$, $C=6$.

$$B^2-AC=4-6=-2<0 .$$

The PDE is **elliptic** (option B). (It is also linear, since $u$ and its derivatives appear only to the first power, but "linear" is a different classification; the question asks for the type.)

Official GATE 2021 answer key: https://gate2026.iitg.ac.in/doc/download/Answer_keys2021/ae_2021.pdf