# Ordinary differential equation

The behavior of a system of ODEs can be visualized through the use of a phase portrait.

Symmetry methods have been applied to differential equations that arise in mathematics, physics, engineering, and other disciplines.

There are several theorems that establish existence and uniqueness of solutions to initial value problems involving ODEs both locally and globally. The two main theorems are

In their basic form both of these theorems only guarantee local results, though the latter can be extended to give a global result, for example, if the conditions of Grönwall's inequality are met.

which is one of the two possible cases according to the above theorem.

Differential equations can usually be solved more easily if the order of the equation can be reduced.

can be written as a system of *n* first-order differential equations by defining a new family of unknown functions

Some differential equations have solutions that can be written in an exact and closed form. Several important classes are given here.

In the case of a first order ODE that is non-homogeneous we need to first find a DE solution to the homogeneous portion of the DE, otherwise known as the characteristic equation, and then find a solution to the entire non-homogeneous equation by guessing. Finally, we add both of these solutions together to obtain the total solution to the ODE, that is: