Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 14 / c / ii / Solution

For $p\ne 0,\pm 1$, separate variables and use partial fraction decomposition:Integration yieldsOn any connected solution interval, absorb the fixed signs into a real constant to writeThe two real signs of the square root are chosen consistently on a branch where $C/(x+C)>0$. The constant solutions of the reduced equation are $p=0,1,-1$. The original equation excludes the last two; $p=0$ is retained by $C=0$ for $x\ne 0$.