As I understand it, the differential equation
$y^{(n)} + p_1(t)y^{(n-1)} + ... + p_{(n-1)}(t)y^{(1)} = f$
is linear because the left hand side can be written as $L[y]$ where $L$ is a linear operator. So why isn't
$p_0(t)y^{(n)} + p_1(t)y^{(n-1)} + ... + p_{n-1}(t)y^{(1)} = f$
considered linear?