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GATE 2017 ME (ME1) – Question 13

Engineering Mathematics · Differential Equations: Heat, wave and Laplace equations · 1 mark · Multiple choice

For a constant $c>1$, the PDE for $u(x,y)$ is $\partial u/\partial y+c\,\partial u/\partial x=0$. Its solution is

  1. $u=f(x+cy)$
  2. $u=f(x-cy)$
  3. $u=f(cx+y)$
  4. $u=f(cx-y)$

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Show answer and explanation

Correct answer: (B) $u=f(x-cy)$

Explanation

Along a characteristic, $dx/dy=c$, giving invariant $x-cy$. Therefore $u=f(x-cy)$ for an arbitrary differentiable function f. Direct differentiation gives $u_y=-cf\prime$ and $u_x=f\prime$, so $u_y+cu_x=0$. Select B.

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