We consider a differentiable map $f$ from an open interval $I$ 

to a uniformly closed linear subspace $A$ of $C(X)$, the Banach 

space of all complex-valued bounded continuous functions 

on a topological space $X$. Let $\varepsilon$ be a non-negative 

real number, $\lambda$ a complex number so that ${\rm Re}\, 

\lambda \neq 0$. Then we show that $f$ can be approximated 

by the solution to $A$-valued differential equation 

$x^{\, '}(t) = \lambda x(t)$, if $\Vert f^{\, '}(t) - 

\lambda f(t) \Vert_{\infty} \leq \varepsilon$ holds for 

every $t \in I$.