In an equation involving \(n\) arbitrary constants, a differential equation of \({n^{{\rm{th}}}}\) order can be obtained by eliminating these \(n\) arbitrary constants. For example, suppose \[y = A\cos x + B\sin x\]then \[y' =  - A\sin x + B\cos x\]  Differentiating again, we have, \[y'' =  - A\cos x - B\sin x =  - (A\cos x + B\sin x)\]

or \(y'' + y = 0\), this is the required differential equation.

Example: Form the differential equation of the family of curves \(y = {c_1}{e^x} + {c_2}{e^{2x}}\)

Solution: The equation has 2 arbitrary constants; hence its order should be 2.

Differentiating the equation, we have,

\[\frac{{dy}}{{dx}} = y' = {c_1}{e^x} + 2{c_2}{e^{2x}}\;.....\;(1)\]

Differentiating again,

\[\frac{{{d^2}y}}{{d{x^2}}} = y'' = {c_1}{e^x} + 4{c_2}{e^{2x}}\;.....(2)\]

From the equations (1) and (2), we have

\[2{c_2}{e^{2x}} = y'' - y'\]\[{c_1}{e^x} = 2y' - y''\]

Putting these values in the given differential equation, we have

\[y = [2y' - y''] + \left[ {\frac{{y'' - y'}}{2}} \right]\]

\[ \Rightarrow y'' - 3y' + 2y = 0\]