Question

Form the differential equation from the following primitive where constants are arbitrary:
y = ax² + bx + c

Answer 1

The equation of family of curves is \[y = a x^2 + bx + c\]                                         ...(1)
where \[a, b\text{ and }c\] are arbitrary constants. So, we shall get a differential equation of third order.
Differentiating equation (1) with respect to x, we get
\[\frac{dy}{dx} = 2ax + b\]                                             ...(2)
Differentiating equation (2) with respect to x, we get

\[\frac{d^2 y}{d x^2} = 2a\]                                           ...(3)
Differentiating equation (3) with respect to x, we get

\[\frac{d^3 y}{d x^3} = 0\]
It is the required differential equation.