Differential equation whose solution is \[y = cx + c - {c^3}\], is
A. \[\frac{{dy}}{{dx}} = c\]
В. \[y = x\frac{{dy}}{{dx}} + \frac{{dy}}{{dx}} - {\left( {\frac{{dy}}{{dx}}} \right)^3}\]
C. \[\frac{{dy}}{{dx}} = c - 3{c^2}\]
D. None of these

Hint:
A function that can be used to anticipate the behavior of the original system, at least under some restrictions, is produced by the solution of the differential equation. In this case, the solution is given, the corresponding differential equation for the given solution to be found. This can be determined by differentiating the given solution with respect to x, where the constant c should be eliminated. And hence we obtained the required differential equation.
Formula used:
\[y = {x^n}\]
Differentiate with respect to x
\[\frac{{dy}}{{dx}} = n{x^{n-1}}\]
Complete step-by-step solution:
We have been given the solution for the differential equation as
\[y = cx + c - {c^3} \ldots \ldots (1)\]
Here \[{\rm{c}}\] is a constant which has to be eliminated.
Now, on differentiating the equation (1) with respect to \[x\], we obtain
\[\frac{{{\rm{d}}y}}{{\;{\rm{d}}x}} = {\rm{c}} \ldots \ldots ..(2)\]
Using the equation (2) in equation (1) we get,
\[y = \left( {\frac{{{\rm{d}}y}}{{\;{\rm{d}}x}}} \right)x + \frac{{{\rm{d}}y}}{{\;{\rm{d}}x}} - {\left( {\frac{{{\rm{d}}y}}{{\;{\rm{d}}x}}} \right)^3}\]
It is the required differential equation.
Therefore, the differential equation for the solution \[y = cx + c - {c^3}\] is \[y = \left( {\frac{{{\rm{d}}y}}{{\;{\rm{d}}x}}} \right)x + \frac{{{\rm{d}}y}}{{\;{\rm{d}}x}} - {\left( {\frac{{{\rm{d}}y}}{{\;{\rm{d}}x}}} \right)^3}\]
Hence, the option B is correct.

Note:
Student should be aware of the connection between the variables x and y that is obtained after removing the derivatives (i.e., integration), where the relation incorporates arbitrary constants to express the order of an equation, is the general solution of the differential equation. In contrast to the second-order differential equation, the first-order differential equation's solution only comprises one arbitrary constant. The general solution of the differential equations is achieved if specific values are assigned to the arbitrary constant. There are some standard forms that can be used to obtain the general solution to the first-order differential equation of the first degree.