Fourier transformation and stability of differential equation on $L^1(\mathbb{R})$

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