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The difference between ordinary differential equations, which we often refer to as ODEs, and partial differential equations, which we often refer to as PDEs, is that ODEs have one independent variable and PDEs have more than one. The theory of ODEs is very well worked out. PDEs are much harder to work with and a lot of research is still being done to determine when a PDE can be solved, when the solution is unique, and to find numerical (computer) methods for approximating solutions.
Partial differential equations are classified into one of three types: parabolic, elliptic, or hyperbolic. The classic example of a parabolic equation is the heat equation. The classic example of an elliptic equation is the Laplace equation. The classic example of a hyperbolic equation is the wave ...
This solution highlights the difference between ordinary differential equations (ODEs), and partial differential equations, (PDEs).