= Singular endpoint compatibility for a differential equation
{title2=$[t(y'+ay)]'=h(t)\implies y'(0)+ay(0)=h(0)$}
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 $[t(y'+ay)]'=h(t)$ has finite continuous $y,y'$ at zero. Integration gives
$$
t(y'+ay)=C+\int_0^t h(s)\,ds.
$$
Regularity forces $C=0$. Dividing by $t>0$ and taking the limit gives $y'(0)+ay(0)=h(0)$, 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 $y'+y=t/2$ and $y=Ae^{-t}+(t-1)/2$. This explains the reduced constant count without treating the singular endpoint as an ordinary second-order initial point.
Back to article page
