In a differential equation, when the variables and their derivatives are only multiplied by constants, then the equation is linear. The variables and their derivatives must always appear as a simple first power.

Why are differential equations linear?

A differential equation is an equation involving derivatives. ... The first two are called linear differential equations because they are linear in the variable y, the first has an "inhomogeneous term" that is independent of y on the right, the second is a homogeneous linear equation since all terms are linear in y.

Are differential operators linear?

We think of the differential operator as operating on functions (that are sufficiently differentiable). The differential operator is linear, that is, for all sufficiently differentiable functions and and all scalars . ... Usually the functions are called eigenfunctions.