Question

Obtain a differential equation of the family of curves $y=a\sin (ax+b)$ where a and b being arbitrary constant.

Answer 1

The numbers of constants is equal to the number of time we differentiate.

Here, there are two constants, so we differentiate twice.

$y=a\sin (ax+b)$

$\Rightarrow \frac{dy}{dx}=a\cos (ax+b)$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=-a^2\sin (ax+b)$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=-ay$

$\Rightarrow \frac{d^{2}y}{d{x}^2}+ay=0$

Hence $\frac{d^{2}y}{d{x}^2}+ay=0$ is the required differential equation.