Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 17 / a

Integrating $y^{\prime \prime }=0$ once gives a constant first derivative, and integrating again givesThere are two independent constants, as expected for a second-order ordinary differential equation.

After $n$ integrations, the general solution is an arbitrary polynomial of degree at most $n-1$:Conversely, every such polynomial has zero $n$th derivative. The $n$ free coefficients span the full solution space on a connected interval.