A linear differential equation (of the second order) is of the form
$a(x)\frac{d^2y}{dx^2}+b(x)\frac{dy}{dx}+c(x)y=d(x).$
$y$ and its derivatives may not appear with a power or as the argument of a function.
Only $1.4$ fulfills this (with $a(x)=x^2,b(x)=0,c(x)=2,d(x)=2$).
The rule generalizes to all orders.
As you can check, with $z=\lambda y$ where $y$ is a solution,
$a(x)\frac{d^2z}{dx^2}+b(x)\frac{dz}{dx}+c(x)z=\lambda a(x)\frac{d^2y}{dx^2}+\lambda b(x)\frac{dy}{dx}+\lambda c(x)y=\lambda d(x),$
and with $z=y_1+y_2$ where $y_1,y_2$ are two solutions,
$a(x)\frac{d^2z}{dx^2}+b(x)\frac{dz}{dx}+c(x)z\\ =a(x)\frac{d^2y_1}{dx^2}+b(x)\frac{dy_1}{dx}+c(x)y_1+a(x)\frac{d^2y_2}{dx^2}+b(x)\frac{dy_2}{dx}+c(x)y_2=2d(x),$
so that in both cases $z$ is also a solution (provided you account for the factors in the RHS).
$1.5$ isn't linear because $z^2=(\lambda y)^2=\lambda^2y^2\ne\lambda y^2$.
$1.6$ isn't linear because $\sin z=\sin\lambda y\ne\lambda\sin y$.