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# Ordinary Differential Equations

Ordinary differential equation initial value problem solvers

The Ordinary Differential Equation (ODE) solvers in MATLAB® solve initial value problems with a variety of properties. The solvers can work on stiff or nonstiff problems, problems with a mass matrix, differential algebraic equations (DAEs), or fully implicit problems. For more information, see Choose an ODE Solver.

## Functions

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 `ode45` Solve nonstiff differential equations — medium order method `ode23` Solve nonstiff differential equations — low order method `ode113` Solve nonstiff differential equations — variable order method
 `ode15s` Solve stiff differential equations and DAEs — variable order method `ode23s` Solve stiff differential equations — low order method `ode23t` Solve moderately stiff ODEs and DAEs — trapezoidal rule `ode23tb` Solve stiff differential equations — trapezoidal rule + backward differentiation formula
 `ode15i` Solve fully implicit differential equations — variable order method `decic` Compute consistent initial conditions for `ode15i`
 `odeget` Extract ODE option values `odeset` Create or modify options structure for ODE and PDE solvers
 `deval` Evaluate differential equation solution structure `odextend` Extend solution to ODE

## Topics

Choose an ODE Solver

ODE background information, solver descriptions, algorithms, and example summary.

Summary of ODE Options

Usage of `odeset` and table indicating which options work with each ODE solver.

ODE Event Location

Detect events during solution of ODE.

Solve Nonstiff ODEs

This page contains two examples of solving nonstiff ordinary differential equations using `ode45`.

Solve Stiff ODEs

This page contains two examples of solving stiff ordinary differential equations using `ode15s`.

Solve Differential Algebraic Equations (DAEs)

Solve ODEs with a singular mass matrix.

Nonnegative ODE Solution

This topic shows how to constrain the solution of an ODE to be nonnegative.

Solve System of ODEs with Multiple Initial Conditions

This example compares two techniques to solve a system of ordinary differential equations with multiple sets of initial conditions.

Troubleshoot Common ODE Problems

FAQ containing common problems and solutions.

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