second-order differential equation

A second-order differential equation is a mathematical statement that relates a function of one variable to its second-order derivatives. The order of a differential equation is determined by the highest order of derivative it contains. For example, the equation ((f'')^2 + (f''')^4 + f = x) is a second-degree, third-order differential equation.

Differential equations are crucial in many scientific and engineering fields because they describe the rates of change of continuously varying quantities. Solving a differential equation yields a function that predicts the system's behavior. While some simple equations can be solved by inspection, many require specialized techniques, and numerical methods are often used to find approximate solutions when explicit solutions are not possible.12

Ordinary differential equations involve functions of a single variable, whereas partial differential equations involve functions of multiple variables and their partial derivatives.13