Question

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

Answer 1

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

Differentiating w.r.t. x, we get,

$\frac{dy}{dx}=a.\cos (x+b).1$

Differentiating w.r.t. x, we get,

$\frac{d^{2}y}{d{x}^2}=a.(-\sin (x+b\left)\right),1$

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

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