In Ordinary Differential Equations (ODEs), we are accustomed to finding a general solution characterized by a specific number of arbitrary constants. However, when we transition to Partial Differential Equations (PDEs), the landscape of solutions becomes significantly more complex. Because PDEs involve multiple independent variables, the degrees of freedom are not just represented by constants, but by arbitrary functions. This page explores the hierarchy of solutions—from the "Complete Integral" (families of surfaces) to the "General Integral" (functional relationships) and the "Particular Integral" (specific physical applications).

A solution of a first-order PDE $F(x, y, z, p, q) = 0$ that contains as many arbitrary constants as there are independent variables is known as a Complete Integral. For a PDE with two independent variables $x$ and $y$, the complete integral is represented by the relationship: $$\begin{equation} \phi(x, y, z, a, b) = 0 \nonumber \end{equation}$$ where $a$ and $b$ are the arbitrary constants. Geometrically, this represents a two-parameter family of surfaces in three-dimensional space.