Does linearity in differential equations only depend on the dependent variable?

Question

This is the example question-

$(1-x)y’’ - 4xy’ + 5y = \cos x$

why is this a linear D.E. ? cosine is not linear so how can it be linear ?

Answer 1

A linear differential equation has the form

$$ a(x)\frac{d^2}{dx^2}y(x) + b(x)\frac{d}{dx}y(x) + c(x)y(x) = f(x) $$

It's linear in the sense that all the operators acting on $y$ are linear operators.

It does not matter whether any of the coefficient functions of $x$ are linear w.r.t $x$.