Q106 / 170 · Manipal (MET) 2021 · Mathematics
Differential equations of the family of curves $y=a \cos \mu x+b \sin \mu x$, where $a$ and $b$ are arbitrary constants is given by
- A$\frac{d^2 y}{d x^2}+\mu y=0$
- B$\frac{d^2 y}{d x^2}-\mu^2 y=0$
- Correct answerC$\frac{d^2 y}{d x^2}+\mu^2 y=0$
- DNone of these
Solution
Given, $y=a \cos \mu x+b \sin \mu x$...(i)
On differentiating w.r.t. $x$, we get
$\frac{d y}{d x}=-a \mu \sin \mu x+b \mu \cos \mu x$
Again, differentiating, we get
$\frac{d^2 y}{d x^2}=-a \mu^2 \cos \mu x-b^2 \mu \sin \mu x$
$=-\mu^2 y$ [from Eq. (i)]
$\Rightarrow \frac{d^2 y}{d x^2}+\mu^2 y=0$
Hence, the correct answer is option: 'C'.