Q88 / 200 · Manipal (MET) 2011 · Mathematics

If $Z=f(x+a y)+\phi(x-a y)$, then

  • A
    $Z_{x x}=Z_{y y}$
  • B
    $Z_{x x}=a^2 Z_{y y}$
  • Correct answer
    C
    $Z_{y y}=a^2 Z_{x x}$
  • D
    None of these

Solution

Given that, $\begin{aligned} Z & =f(x+a y)+\phi(x-a y) \\ Z_x & =f^{\prime}(x+a y)+\phi(x-a y) \\ \Rightarrow \quad Z_{x x} & =f^{\prime \prime}(x+a y)+\phi^{\prime \prime}(x-a y)\end{aligned}$ and $\quad Z_y=a f^{\prime}(x+a y)-a \phi^{\prime}(x-a y)$ $\Rightarrow \quad z_{y y}=a^2 f^{\prime \prime}(x+a y)-a^2 \phi^{\prime}(x-a y)$ Hence, $a^2 Z_{x x}=Z_{y y}$

Hence, the correct answer is option: 'C'.

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