Find the differential equation of the family of curves, `x=A\ cos n t+Bsinn t ,` where A and B are arbitrary constants.

Explanation

`x=A cos n t+Bsinn t ldots`(1)

Differentiating it with respect to `t`, we get

`{dx}/{dt}= -A nsinn t +Bncos n t `

Differentiating it with respect to `t`, we get

`{d^2x}/{dt^2}= -An^2 cos n t-Bn^2sinn t ldots`(2)

From (1) and (2),

`{d^2y}/{dx^2}+n^2x=0`

This is the required differential equation.