Linear differential operators

A linear differential operator is a mathematical tool that acts on functions, involving derivatives, in a way that respects addition and scalar multiplication. Essentially, it takes a function and produces another function by combining derivatives (rates of change) linearly. For example, an operator like \( L(y) = y'' + 3y' - 2y \) applies rules that preserve linearity: \(L(f + g) = L(f) + L(g)\) and \(L(cf) = cL(f)\). These operators are fundamental in modeling systems where change depends on rates, such as physics and engineering, allowing the formulation and solution of differential equations describing diverse phenomena.