Question

From the differential equation of the family of curves representing $y=a\cos (x+b)$ where a and b are arbitrary constants.

Answer 1

Given $y=a\cos (x+b)$

Differentiating it w.r.t $x$

$\frac{dy}{dx}=-a\sin (x+b)$

Again differentiating it w.r.t $x$

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

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

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

This is the required equation.