# Ordinary Differential Equations (ODE)¶

ODE stands for Ordinary Differential Equation.

An ODEProblem represents a first order ordinary differential equation (ODE) with a given state at an initial time. This ODEProblem is built with a function of time and state, as well as an array describing the intial state, and a time interval.

An ODEResult represents the solution of an ODE evaluated at a discrete set of times within the specified time interval.

Note

This feature is under active development. Future iterations include the integration of ODEProblem s with MDODiscipline.

## Architecture¶

### ODEProblem and ODEResult¶

The main classes in the ODE submodule are the ODEProblem and ODEResult. These represent respectively the first-order ODE with its initial conditions, and the solution of this problem evaluated at a discrete set of values for time.

As a reminder, a first-order ordinary differential equation is an equation of the form:

$\frac{ds}{dt}(t) = f(t, s(t)) \ \textrm{ and }\ s(t_0) = s_0$

where $$s$$ is the state which depends on $$t$$, the time. The right-hand side function $$f$$ is a function of the time and the state. The value of the state at an initial time $$t_0$$ is known to be $$s_0$$.

The solution of this problem is provided for discrete values of time within a given interval $$[t_0,\ t_f]$$.

### Classes¶

The classes described by the ODE module are as such:

### Packages¶

The submodules are organized in the following fashion.