Chapter 8 - Linear Differential Equations of Order n - 8.1 General Theory for Linear Differential Equations - Problems - Page 503: 7

0; $y (x)= (\sin x^2)\in ker (L)$

We have: $Ly=(D^2-x^{-1}D+4x^2)(\sin x^2)$ $L(\sin x^2)=2(\cos x^2)-4x^2(\sin x^2)-x^{-1}\times 2x\cos x^2+4x(\sin x^2)\\=2 \cos x^2-2\cos x^2-4x^2\sin x^2+4x^2\sin x^2\\=0$ This yields $y (x)= (\sin x^2)\in ker (L)$