Singular endpoint compatibility for a differential equation ($[t(y^{\prime }+ay){]}^{\prime }=h(t)⟹y^{\prime }(0)+ay(0)=h(0)$)

A vanishing highest-derivative coefficient can make the two initial values of a second-order differential equation dependent. Suppose $h$ is continuous at zero, $a$ is constant, and a solution of $[t(y^{\prime }+ay){]}^{\prime }=h(t)$ has finite continuous $y,y^{\prime }$ at zero. Integration givesRegularity forces $C=0$. Dividing by $t>0$ and taking the limit gives $y^{\prime }(0)+ay(0)=h(0)$, since the integral average tends to $h(0)$. The remaining equation is first order and therefore has only one free regular initial value. For $a=1$ and $h(t)=t$, it yields $y^{\prime }+y=t/2$ and $y=A{e}^{-t}+(t-1)/2$. This explains the reduced constant count without treating the singular endpoint as an ordinary second-order initial point.