The purpose of this article is to establish a kind of stricter Hyers–Ulam stability of second order linear differential equation of Carathéodory type. More explicitly, we prove that if $x$ is an approximate solution satisfying a kind of stricter condition of the differential equation $x^{\prime \prime }\left(t\right)+b_1{x}^{\prime }\left(t\right)+b_{0}x\left(t\right)=f\left(t\right)$ without the assumption of continuity of $f\left(t\right)$, then there exists an exact solution of the differential equation near to $x$.

中文翻译：

---

Carathéodory型二阶线性微分方程的一种更严格的Hyers-Ulam稳定性

本文的目的是建立一种更严格的Carathéodory型二阶线性微分方程的Hyers-Ulam稳定性。更明确地说，我们证明$X$ 是满足微分方程一种更严格条件的近似解 $X^{\prime \prime }（Ť）+b_{\mathrm{1个}}{X}^{\prime }（Ť）+b_{0}X（Ť）=F（Ť）$ 没有假设的连续性 $F（Ť）$，那么在附近有一个微分方程的精确解 $X$。