I am getting confused on differential operators, can some please help explain the following. Suppose I have two differential operators $B = D_x^2 + \alpha A I$ where $I$ is the identity operator and $C = uI $ where $u(x,t)$. I would like to compute $L = BC$ for which the answer is $L = u_{xx}I+2u_xD_x+uD_x^2+αAuI$ but I am not sure how B operates on C?
Here is my attempt:
$(D_x^2 + \alpha A I)(uI) = D_x^2uI + \alpha A I uI $
which then I simplify to:
$u_{xx}I + \alpha A u I$
which is clearly wrong
Thanks for any pointers