Chapter 8 - Linear Differential Equations of Order n - 8.1 General Theory for Linear Differential Equations - Problems - Page 503: 8

$y (x)= (\sin x+\cos x) \in ker (L)$

We have: $Ly=(D^3+D^2+D+1)(\sin x+\cos x)$ $L(\sin x+\cos x)=(-\cos x+\sin x)+(-\sin x-\cos x)+(\cos x -\sin x)+(\cos x -\sin x)\\=-\cos x+\sin x -\sin x -\cos x+\cos x-\sin x+\cos x +\sin x \\=0$ This yields $y (x)= (\sin x+\cos x) \in ker (L)$