Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 2 / 9 / d / Solution

Differentiate $z^{\prime }=yz$ without dividing by $z$:Thus the linearization of a Riccati equation givesIts characteristic polynomial is $(\lambda -a)(\lambda -b)$, explaining both the distinct-root exponentials and the repeated-root factor $A+Bx$. Nonzero solutions reconstruct $y=z^{\prime }/z$ between their zeros; the identically zero $z$ also satisfies the linear equation.