 Solution

The zero initial value and slope select a forward Green's function: set it to zero for $x<\xi$, and choose the right piece to start at zero with slope one. This givesFor $f=1$ on $(0,1)$ and zero elsewhere,Indeed $y_2={\int }_0^{\min (x,1)}(x-{\xi }^2/x)d\xi /2$. It and its first derivative match at one, while $y_2,y_2^{\prime }\to 0$ at zero. The only homogeneous solution with these two initial limits is zero, so this solution too is unique. Its growth at infinity is allowed by these boundary conditions.