Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 2 / 6A / b / Solution

For two solutions $y_1,y_2$, define their Wronskian byDifferentiation and substitution of the second-order linear differential equation giveThis proves the Abel identity and hence

On an interval where the known solution $y_1$ does not vanish,Integrating gives the reduction of order formulaChoosing the normalization $C=1$ makes the Wronskian nonzero, so this solution is linearly independent of $y_1$. To verify it really solves the equation, write $y_2=y_{1}h$. Its equation reduces tothe integrand gives $h^{\prime }=\exp (-\int p)/y_1^2$, which satisfies this equation on differentiation. Adding an integration constant only adds a multiple of $y_1$. The formula is local where $y_1\ne 0$; at a zero of $y_1$ with regular coefficients, the actual solution can be continued by initial-value existence and uniqueness, even if its integral representation has an apparent singularity.