Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 8 / c / Solution

Define the Wronskian by $W=y_1{y}_2^{\prime }-y_1^{\prime }{y}_2$. Differentiating and using the normal-form differential equation givesIntegrating this first-order linear differential equation proves Abel's identity:Thus, on any interval avoiding zero and $m$,Absolute-value signs from real logarithms can be absorbed into the constant on each such interval. For linearly independent solutions, $C\ne 0$.