Proposition
26
[1SR]Given parameters \(y_ 0,\ldots ,y_{n-1}ββ\), and also \(πΌββ\), the solution of the Cauchy problem
\[ \begin{cases} p(D) f = e^{πΌ x} \\ f(0)=y_ 0,\\ \ldots \\ f^{n-1}(0)=y_{n-1} \end{cases} \]
exists for all times, and depends continuously on the parameters \(πΌ,y_ 0,\ldots ,y_{n-1}ββ\).