 Solution

Differentiate once less than might seem necessary: writing $v=h^{\prime }$ turns the equation intoUniqueness for this first-order linear equation also covers the solution $v\equiv 0$. Thus every solution has $h^{\prime }(z)=C[z(z-1){]}^{-4/\kappa }$. Since $4/\kappa <1$, the derivative is integrable at $1$. The specified boundary value fixes the additive constant, givingThe strict positivity forces $C>0$; $C=0$ gives the zero function. These are all the positive solutions. The function $s_{\kappa }$ is the increasing scale function of a one-dimensional diffusion for the ratio process.