# 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) `ODEResults` Results of ODE integration (Since R2023b)

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

## Functions

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 `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
 `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