Question

Form the differential equation from the following primitive where constants are arbitrary:
y = cx + 2c² + c³

Answer 1

The equation of family of curves is \[y = cx + 2 c^2 + c^3..............(1)\]

Where `c` is an arbitrary constant.

This equation contains only one arbitrary constant, so we shall get a differential equation of first order.

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

\[\frac{dy}{dx} = c...............(2)\]

Putting the value of `c`  in equation (1), we get

\[y = x\frac{dy}{dx} + 2 \left( \frac{dy}{dx} \right)^2 + \left( \frac{dy}{dx} \right)^3\]

It is the required differential equation.