the linear differential operator

So far you have read a differential equation as a relationship to be untangled. The operator viewpoint offers a cleaner mental move: treat 'differentiate' as a machine, a verb you can name, and then the whole left side of an equation becomes a single machine applied to y. Feed a function in, get a function out. This little shift in framing turns out to unlock a powerful algebra.

Start with the basic machine D, defined by D y = y' — D simply means 'take the derivative.' Apply it twice and you get D^2 y = y''; in general D^k y = y^(k). Now bundle: the linear differential operator L = a_n D^n + a_(n-1) D^(n-1) + ... + a_1 D + a_0 is the single machine that does the whole left side at once, so the equation a_n y^(n) + ... + a_0 y = g becomes simply L[y] = g. The word 'linear' captures the two rules that make L tame: L[y1 + y2] = L[y1] + L[y2] and L[c y] = c L[y]. Those are exactly the rules that power superposition and let you split a forced problem into homogeneous plus particular.

Why is this worth the renaming? Because once L is an object, you can do algebra with it: when the coefficients are constant you can multiply, add, and even factor these operators almost like polynomials in D, which underlies operator factorization and the annihilator method. The honest caution is about ORDER OF OPERATIONS: for constant-coefficient operators the algebra is fully polynomial-like and they commute, but for VARIABLE coefficients operators generally do NOT commute — D and the operator 'multiply by x' satisfy D x - x D = 1, not zero — so you cannot freely reorder factors there.