Consider the linear ODE $$a\left(t\right)\dot{x}+b\left(t\right)x+c\left(t\right)=0$$ with $a,b,c$ are smooth real-valued functions defined on some open interval $I\subset\mathbb{R}$. There is unique $t_s$ $$\exists \, ! \, t_{s} \, \in I: a\left(t_{s}\right)=0$$ While the functions $b\left(t_{s}\right)\ne0, c\left(t_{s}\right)\ne0$.
I need solution $x\left(t\right)$ of the DE with $x\left(t_{1}\right)=0$ (for some $t_1 \in I \backslash \{t_s\}$) to be a smooth function on $I$.
Question: what are the requirements on function a,b,c, and value $t_1$? Are there some common theorems?