Let $X$ be a solution to the boundary value problem $$ X^{\prime\prime}(s) = A(s)X(s), \quad X(0) = X_0, ~~X(t) = X_1,$$ where $0 < t \leq 1$, $A$ is some matrix-valued function, defined on $[0, 1]$, and $X_0$, $X_1$ are certain fixed matrices (suppose the problem has a unique solution for all $0<t \leq 1$.)

Does there exist a constant $C>0$ independent of $ t\in (0, 1]$ such that $|X(s)| \leq C$ for all $s \in [0, t]$? If yes, how does the constant depend on $A$?