Let L be the linear differential operator of order 2 given by L = D2+5xD−3. Compute L(x).

Question

I have no idea where to even begin with question, and my book isn't very helpful on this topic.

Answer 1

I guess what they mean by $L$ is the operator \begin{equation} L = \frac{\partial^2}{\partial x^2} + 5 x \frac{\partial}{\partial x} -3 \ . \end{equation} Then if we apply this to $x$ we have \begin{eqnarray} L(x) &=& \frac{\partial^2}{\partial x^2} x + 5 x \frac{\partial}{\partial x} x -3 x \ , \\ &=& 0 + 5x - 3x\ , \\ &=& 2x \ . \end{eqnarray}