Chapter 8 - Linear Differential Equations of Order n - 8.1 General Theory for Linear Differential Equations - Problems - Page 503: 5

$y \in ker(L)$

We have: $Ly=(D^2-4D+4)(xe^{2x})$ $Ly=D^2(xe^{2x})-4D(xe^{2x})+4(xe^{2x})\\=D(e^{2x})+2xe^{2x})-4(e^{2x}+2xe^{2x})+4x e^{2x}\\=2e^{2x}+2e^{2x}+4xe^{2x}-4e^{2x}-8xe^{2x}+4xe^{2x}\\=0$ This yields $y \in ker(L)$