Real world problems in general, involve functions of several (independent) variables giving rise to partial differential equations more frequently than ordinary differential equations. Thus, most problems in engineering and science reproduce with first and second order linear non-homogeneous partial differential equations.
Partial differential equations (PDEs) are equations that involve rates of change with respect to continuous variables. The position of a rigid body is specified by six numbers, but the configuration of a fluid is given by the continuous distribution of several parameters, such as the temperature, pressure, and so forth. The dynamics for the rigid body take place in a finite-dimensional configuration space; the dynamics for the fluid occur in an infinite-dimensional configuration space. This distinction usually makes PDEs much harder to solve than ordinary differential equations (ODEs), but here again there will be simple solutions for linear problems. Classic domains where PDEs are used include acoustics, fluid flow, acoustics, fluid flow, electrodynamics, and heat transfer.