ODEs that are linear
involve only state variables
derivatives of the state variables
constants times those things
and constants by themselves.
No powers, no product,
and no transcendentals.
The first equation has a power in it
so it is nonlinear.
The second equation has
only state variables
and constants times state variables
and derivatives of state variables in it.
So that is a linear ODE.
The third one has a product
of a state variable and a derivative
of a state variable.
Which makes it nonlinear
and the fourth one has a transcendental
function in it, making it too nonlinear