Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2008 / Paper 1 / 3 / i / Solution

The factor satisfying the left condition is $x$, while $x^{-1}$ satisfies decay at infinity. Matching the value and unit derivative jump givesFor the specified compactly supported forcing, integrating over $0<\xi <1$ givesFor example, on $0<x<1$, the two pieces contribute $-x^2/6$ and $-x(1-x)/2$. Both $y_1$ and $y_1^{\prime }$ match at one, and the function tends to zero at both ends. A homogeneous difference $ax+b/x$ with those two limits must vanish, establishing uniqueness.