why does 1/4 wavelength have to have a ground plane and 1/2 doesn't?

I know that an antenna that has half a wavelength does not need the ground plane (example a dipole antenna) and when using a wavelength of 1/4 it is usually necessary to have the ground plane (planar antennas [PCB]). My problem: I can't understand why we need to use a ground plane at 1/4 wavelength and 1/2 wavelength not needed.

Please could also provide source of relevant information, examples, books or scientific article, something like that.
Regarding the subject

 

https://en.wikipedia.org/wiki/Image_antenna
https://en.wikipedia.org/wiki/Ground_plane#Radio_antenna_theory

The radio waves from an antenna element that reflect off a ground plane appear to come from a mirror image of the antenna located on the other side of the ground plane. In a monopole antenna, the radiation pattern of the monopole plus the virtual "image antenna" make it appear as a two element center-fed dipole antenna. So a monopole mounted over an ideal ground plane has a radiation pattern identical to a dipole antenna. The feedline from the transmitter or receiver is connected between the bottom end of the monopole element and the ground plane. The ground plane must have good conductivity; any resistance in the ground plane is in series with the antenna, and serves to dissipate power from the transmitter.

 

The differential equations cannot tell the difference between the two cases. The can certainly tell if the 1/4 wave vertical has no ground plane.