How do ordinary differential equations differ from partial differential equations?

In mathematics, the primary distinction between ordinary differential equations (ODEs) and partial differential equations (PDEs) lies in the number of independent variables upon which the unknown function depends.12

• Ordinary Differential Equations (ODEs): These involve a function that depends on only a single independent variable. Consequently, the equation contains only ordinary derivatives, which represent the rate of change of the function with respect to that one variable.12
• Partial Differential Equations (PDEs): These involve a function that depends on several independent variables. As a result, the equation contains partial derivatives, which describe how the function changes with respect to each of those variables individually.12

Because of these fundamental differences in structure, the mathematical theories used to study and solve ODEs and PDEs are distinct and are typically treated separately. In the field of numerical analysis, both types of equations are frequently solved using approximation methods—such as finite difference or finite element techniques—to obtain useful solutions for complex systems in science and engineering.23