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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Differential Equations

NDA 2018 · Mathematics · Differential Equations

The differential equation of the family of curves y=p (a x)+q (ax)...

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

Options

  1. Ad2ydx2+ay=0\frac{d^2 y}{d x^2}+a y=0
  2. Bd2ydx2−ay=0\frac{d^2 y}{d x^2}-a y=0
  3. Cd2ydx2+\frac{d^2 y}{d x^2}+ ay =0=0
  4. Dd2ydx2+a2y=0\frac{d^2 y}{d x^2}+a^2 y=0

Correct answer

D. d2ydx2+a2y=0\frac{d^2 y}{d x^2}+a^2 y=0

Step-by-step solution

y=pcos⁡ax+qsin⁡ax⇒dydx=−pasin⁡ax+qacos⁡ax⇒d2ydx2=−pa2cos⁡ax−qa2sin⁡ax=−a2y⇒d2ydx2+a2y=0\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}
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