Let D be an operator defined by
Note: An operator maps a function to another function. (or vector space to another vector space) ie.
Consider the following first order linear differential equation.
(defined to be so)
Consider the second order linear differential equation.
(apply partial fractions ridiculously with complex numbers)
Does one need to understand your digestive system before you take your dinner? – Heaviside
The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture.