Past exam of the year 1 of the computer science course of the University of Oxford / 2005 / Calculus, linear algebra, discrete mathematics, logic and proof, and models of computation / 3 / a / ii / Solution

For $y_1=e^x/x$, on an interval avoiding zero,Substitution into the ordinary differential equation givesTo find an independent solution, put $v=xy$. The product rule gives $v^{\prime \prime }=x{y}^{\prime \prime }+2y^{\prime }$, so the ordinary differential equation becomes $v^{\prime \prime }-v=0$. Its independent solutions are $e^x$ and $e^{-x}$. HenceThe two displayed solutions are linearly independent, since their ratio $e^{2x}$ is nonconstant on any open interval. Constants can be chosen separately on the two sides of the singular point $x=0$.