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