  In the present paper by the Fourier transform we show that every linear
differential equations of $n$-th order has a solution in $L^1(\Bbb{R})$ which
is infinitely differentiable in $\Bbb{R} \setminus \{0\}$. Moreover the
Hyers-Ulam stability of such equations on $L^1(\Bbb{R})$ is investigated.
