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