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

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Solution

The equation of the family of curves is
$x = A \cos n t + B \sin n t$ ...(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
$\frac{d x}{d t} = - A n \sin n t + B n \cos n t$ ...(2)
Differentiating equation (2) with respect to $t$ , we get
$\frac{d^{2} x}{d t^{2}} = - A n^{2} \cos n t - B n^{2} \sin n t \Rightarrow \frac{d^{2} x}{d t^{2}} = - n^{2} (A \cos n t + B \sin n t) \Rightarrow \frac{d^{2} x}{d t^{2}} = - n^{2} x \Rightarrow \frac{d^{2} x}{d t^{2}} + n^{2} x = 0 It is the required differential equation .$

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