section_path: Differential equations/Ordinary differential equations
section_name: Ordinary differential equations
paragraph_text: Ordinary differential equations
  
  An ordinary differential equation or ODE is an equation containing a function of one independent variable and its derivatives. The term " ordinary " is used in contrast with the term partial differential equation, which may be with respect to more than one independent variable.
  
  Linear differential equations, which have solutions that can be added and multiplied by coefficients, are well-defined and understood, and exact closed-form solutions are obtained. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by elementary functions in closed form: Instead, exact and analytic solutions of ODEs are in series or integral form. Graphical and numerical methods, applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often sufficing in the absence of exact, analytic solutions.