Question

Form the differential equation of the family of curves,

x = A cos nt + B sin nt, where A and B are arbitrary constant.

Answer 1

As the given equation has two different arbitrary constants so we can differentiate it twice with respect to x.

x = A cos nt + B sin nt

On differentiating with respect to t we get,

\(\frac{dx}{dt} = \) -An sin nt + Bn cos nt

Again, differentiating with respect to x,

\(\frac{d^2x}{dt^2}\) = -An² cos nt + Bn² sin nt

\(\Rightarrow\frac{d^2x}{dt^2}=\) -n²(A cos nt + B sin nt)

As x = A cos nt + B sin nt

Hence, \(\frac{d^2x}{dt^2}+n^2x =0\) is the required differential equation.