Question

Form the differential equation representing the family of curves $y=a\sin (x+b)$, where $a,b$ are arbitrary constants.

Answer 1

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

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

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

Now, differentiating both sides,

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

Differentiating again on both sides,

$\Rightarrow$ $\frac{d^{2}y}{d{x}^2}=\frac{d\left(a\cos \right(x+b\left)\right)}{dx}$

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

Using $y=a\sin (x+b)$

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

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

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