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.

Solve problems and set options using a visual interface with the Solve ODE Live Editor task.

Objects

`ode`	Ordinary differential equations (Since R2023b)
`odeMassMatrix`	ODE mass matrix (Since R2023b)
`odeJacobian`	ODE Jacobian matrix (Since R2023b)
`odeEvent`	ODE event definition (Since R2023b)
`odeSensitivity`	ODE sensitivity analysis (Since R2024a)
`odeDelay`	ODE delay definition (Since R2025a)
`ODEResults`	Results of ODE integration (Since R2023b)

Live Editor Tasks

Solve ODE

Solve system of ordinary differential equations in the Live Editor (Since R2024b)

Functions

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Nonstiff Solvers

`ode45`	Solve nonstiff differential equations — medium order method
`ode23`	Solve nonstiff differential equations — low order method
`ode78`	Solve nonstiff differential equations — high order method (Since R2021b)
`ode89`	Solve nonstiff differential equations — high order method (Since R2021b)
`ode113`	Solve nonstiff differential equations — variable order method

Stiff Solvers

`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

Fully Implicit Solvers

`ode15i`	Solve fully implicit differential equations — variable order method
`decic`	Compute consistent initial conditions for `ode15i`

Get/Set Options

`odeget`	Extract ODE option values
`odeset`	Create or modify options structure for ODE and PDE solvers

Evaluate and Extend Solution

`deval`	Evaluate differential equation solution structure
`odextend`	Extend solution to ODE

Topics

Solving ODEs

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.
Troubleshoot Common ODE Problems
FAQ containing common problems and solutions.

Automatic Differentiation in ODEs

Resolve Error: Unable to Assign Expression During Automatic Differentiation
Troubleshoot expression initialization for automatic differentiation.
Supported Operations on Expressions
List of supported operations on expressions for automatic differentiation.
Automatic Differentiation Background
Learn the theoretical aspects of automatic differentiation and its forward and reverse modes.

Featured Examples

Solve Predator-Prey Equations

Solve a differential equation representing a predator/prey model using variable step size Runge-Kutta integration methods. The ode23 method uses a 2nd and 3rd order pair of formulas for medium accuracy, and the ode45 method uses a 4th and 5th order pair for higher accuracy.

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Solve Equations of Motion for Baton Thrown into Air

Solves a system of ordinary differential equations that model the dynamics of a baton thrown into the air [1]. The baton is modeled as two particles with masses m1 and m2 connected by a rod of length L. The baton is thrown into the air and subsequently moves in the vertical xy-plane subject to the force due to gravity. The rod forms an angle θ with the horizontal and the coordinates of the first mass are (x,y). With this formulation, the coordinates of the second mass are (x+Lcosθ,y+Lsinθ).

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Sensitivity Analysis of Epidemic ODE Parameters

Perform sensitivity analysis on the parameters of an ordinary differential equation (ODE) system that models the spread of a disease in an epidemic.

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Solve Celestial Mechanics Problem with High-Order Solvers

Use ode78 and ode89 to solve a celestial mechanics problem that requires high accuracy in each step from the ODE solver for a successful integration. Both ode45 and ode113 are unable to solve the problem using the default error tolerances. Even with more stringent error thresholds, ode89 performs best on the problem due to the high accuracy of the Runge-Kutta formulas it uses in each step.

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Solve Stiff Transistor Differential Algebraic Equation

Solve a stiff differential algebraic equation (DAE) that describes an electrical circuit [1]. The one-transistor amplifier problem coded in the example file amp1dae.m can be rewritten in semi-explicit form, but this example solves it in its original form Mu′=ϕ(u). The problem includes a constant, singular mass matrix M.

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Solve System of ODEs with Multiple Initial Conditions

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

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Solve ODE with Strongly State-Dependent Mass Matrix

Solve Burgers' equation using a moving mesh technique [1]. The problem includes a mass matrix, and options are specified to account for the strong state dependence and sparsity of the mass matrix, making the solution process more efficient.

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