The general solution of a partial differential equation is usually richer than the general solution of an ordinary differential equation. In an ordinary differential equation of order $n$, the general solution generally contains $n$ arbitrary constants. In a partial differential equation, the general solution often contains arbitrary functions. This is one of the main reasons why partial differential equations require a separate theory.

For ordinary differential equations, after the general solution is found, a particular solution is usually obtained by determining arbitrary constants from prescribed conditions. For partial differential equations, the situation is more delicate. Since the general solution may contain arbitrary functions, the prescribed supplementary conditions must determine functions, not merely constants. Thus, in partial differential equations, the passage from a general solution to a particular solution can be as difficult as finding the general solution itself.