Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2003 / Paper 1 / 11 / a / i / Solution

For a nonzero solution, separation of variables gives $dy/y=kdx/(1+x^2)$, so $\ln |y|=k\arctan x+C$. Restoring the zero solution by allowing an arbitrary real multiplicative constant givesIt is valid for all real numbers $x$, including the constant solutions when $k=0$.