Answer
$(Ly)(x)=8x^2e^{2x}+x^2\sin x-2e^{2x}\sin x+\sin^2 x$
Work Step by Step
Given $L=x^2D^3-\sin xD$
with $y_1=e^{2x}+\cos x$
We first compute the appropriate derivatives
$y'(x)=2e^{2x}-\sin x\\
y''(x)=4e^{2x}-\cos x$
Hence
$(Ly)(x)=(e^{2x}+\cos x)(x^2D^3-\sin xD)=8x^2e^{2x}+x^2\sin x-2e^{2x}\sin x+\sin^2 x$
