The motion of electrons in an electric field with cylindrical symmetry is a good example of a two-dimensional problem that is easy to solve using the Feynman algorithm, but difficult to solve by analytic methods. The field can be established by placing conducting plates in the x-z plane: a positively charged plate in the left half-plane, and a negatively charged plate in the right half-plane (with a thin insulating strip between them). Electrons will be attracted to the positive plate and repelled from the negative plate. In the example, a source of electrons is placed on the 45-degree line in the first quadrant of the x-y plane. The source emits monoenergetic electrons in the x-y plane. The energy is such that in order to reach the negative plate an electron would have to convert all of its kinetic energy to potential energy. Four fieldlines are plotted. The direction of the field is tangent to a fieldline, and the magnitude of the field at any point is inversely proportional to the length of a fieldline through that point. All that the two dimensional Feynman algorithm needs to get started is expressions for the x and y components of the force. In the example, electron trajectories are plotted for 12 initial angles.
I first encountered this problem (or one like it) as a graduate student. I was to solve the problem graphically using a compass to draw Huygens' wavelets on a large sheet of graph paper, then I was to plot trajectories perpendicular to the wavefronts. After that I think I was to leap a tall building in a single bound . . .