Question

Find the differential equation of the family of curves, x = A cos nt + B sin nt, where A and B are arbitrary constants.

Answer 1

The equation of the family of curves is \[x = A\cos nt + B\sin nt\]                              .........(1)

where `A" and "B` are arbitrary constants.

This equation contains two arbitrary constants, so we shall get a differential equation of second order.

Differentiating equation (1) with respect to t, we get

\[\frac{dx}{dt} = - \text{ An }\sin\text{ nt }+ \text{ Bn }\cos nt\]                          ............(2)

Differentiating equation (2) with respect to t, we get

\[\frac{d^2 x}{d t^2} = - A n^2 \cos nt - B n^2 \sin nt\]

\[ \Rightarrow \frac{d^2 x}{d t^2} = - n^2 \left( A\cos nt + B\sin nt \right)\]

\[ \Rightarrow \frac{d^2 x}{d t^2} = - n^2 x\]

\[ \Rightarrow \frac{d^2 x}{d t^2} + n^2 x = 0 \]

It is the required differential equation .