Q237 / 270 · NDA 2018 (Phase 2) · Mathematics

The differential equation of the family of curves $y=p \cos (a x)$ $+\mathrm{q} \sin (\mathrm{ax})$, where $\mathrm{p}, \mathrm{q}$ are arbitrary constants, is

  • A
    $\frac{d^2 y}{d x^2}+a y=0$
  • B
    $\frac{d^2 y}{d x^2}-a y=0$
  • C
    $\frac{d^2 y}{d x^2}+$ ay $=0$
  • Correct answer
    D
    $\frac{d^2 y}{d x^2}+a^2 y=0$

Solution

$\begin{aligned} & y=p \cos a x+q \sin a x \\ & \Rightarrow \frac{d y}{d x}=-p a \sin a x+q a \cos a x \\ & \Rightarrow \frac{d^2 y}{d x^2}=-p a^2 \cos a x-q a^2 \sin a x=-a^2 y \\ & \Rightarrow \frac{d^2 y}{d x^2}+a^2 y=0\end{aligned}$

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

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