MATLAB Function Reference 
Define a differential equation problem for ordinary differential equation (ODE) solvers
Note
This reference page describes the odefile and the syntax of the ODE solvers used in MATLAB, Version 5. MATLAB, Version 6, supports the odefile for backward compatibility, however the new solver syntax does not use an ODE file. New functionality is available only with the new syntax. For information about the new syntax, see odeset or any of the ODE solvers.

Description
odefile
is not a command or function. It is a help entry that describes how to create an Mfile defining the system of equations to be solved. This definition is the first step in using any of the MATLAB ODE solvers. In MATLAB documentation, this Mfile is referred to as an odefile
, although you can give your Mfile any name you like.
You can use the odefile
Mfile to define a system of differential equations in one of these forms
The ODE file must accept the arguments t
and y
, although it does not have to use them. By default, the ODE file must return a column vector the same length as y
.
All of the solvers of the ODE suite can solve , except ode23s,
which can only solve problems with constant mass matrices. The ode15s
and ode23t
solvers can solve some differentialalgebraic equations (DAEs) of the form .
Beyond defining a system of differential equations, you can specify an entire initial value problem (IVP) within the ODE
Mfile, eliminating the need to supply time and initial value vectors at the command line (see Examples).
To Use the ODE File Template
help odefile
to display the help entry.
switch flag case '' % Return dy/dt = f(t,y). varargout{1} = f(t,y,p1,p2); case 'init' % Return default [tspan,y0,options]. [varargout{1:3}] = init(p1,p2); case 'jacobian' % Return Jacobian matrix df/dy. varargout{1} = jacobian(t,y,p1,p2); case 'jpattern' % Return sparsity pattern matrix S. varargout{1} = jpattern(t,y,p1,p2); case 'mass' % Return mass matrix. varargout{1} = mass(t,y,p1,p2); case 'events' % Return [value,isterminal,direction]. [varargout{1:3}] = events(t,y,p1,p2); otherwise error(['Unknown flag ''' flag '''.']); end %  function dydt = f(t,y,p1,p2) dydt = < Insert a function of t and/or y, p1, and p2 here. > %  function [tspan,y0,options] = init(p1,p2) tspan = < Insert tspan here. >; y0 = < Insert y0 here. >; options = < Insert options = odeset(...) or [] here. >; %  function dfdy = jacobian(t,y,p1,p2) dfdy = < Insert Jacobian matrix here. >; %  function S = jpattern(t,y,p1,p2) S = < Insert Jacobian matrix sparsity pattern here. >; %  function M = mass(t,y,p1,p2) M = < Insert mass matrix here. >; %  function [value,isterminal,direction] = events(t,y,p1,p2) value = < Insert event function vector here. > isterminal = < Insert logical ISTERMINAL vector here.>; direction = < Insert DIRECTION vector here.>;
Notes
t
and y
vectors from the ODE solvers and must return a column vector the same length as y
. The optional input argument flag
determines the type of output (mass matrix, Jacobian, etc.) returned by the ODE file.
switch
statement determines the type of output required, so that the ODE file can pass the appropriate information to the solver. (See notes 4  9.)
'init'
) case, the ODE file returns basic information (time span, initial conditions, options) to the solver. If you omit this case, you must supply all the basic information on the command line.
'jacobian'
case, the ODE file returns a Jacobian matrix to the solver. You need only provide this case when you want to improve the performance of the stiff solvers ode15s
, ode23s
, ode23t
, and ode23tb
.
'jpattern'
case, the ODE file returns the Jacobian sparsity pattern matrix to the solver. You need to provide this case only when you want to generate sparse Jacobian matrices numerically for a stiff solver.
'mass'
case, the ODE file returns a mass matrix to the solver. You need to provide this case only when you want to solve a system in the form .
'events'
case, the ODE file returns to the solver the values that it needs to perform event location. When the Events
property is set to on
, the ODE solvers examine any elements of the event
vector for transitions to, from, or through zero. If the corresponding element of the logical isterminal
vector is set to 1
, integration will halt when a zerocrossing is detected. The elements of the direction
vector are 1
, 1
, or 0
, specifying that the corresponding event must be decreasing, increasing, or that any crossing is to be detected.
flag
generates an error.
Examples
The van der Pol equation,
, is equivalent to a system of coupled firstorder differential equations.
defines this system of equations (with ).
To solve the van der Pol system on the time interval [0 20]
with initial values (at time 0
) of y(1) = 2
and y(2) = 0
, use
To specify the entire initial value problem (IVP) within the Mfile, rewrite vdp1
as follows.
function[out1,out2,out3]
=
vdp1(t,y,flag) if nargin < 3  isempty(flag) out1
=
[y(1).*(1y(2).^2)y(2);
y(1)]; else switch(flag) case 'init' % Return tspan, y0, and options. out1 = [0 20]; out2 = [2; 0]; out3 = []; otherwise error(['Unknown request ''' flag '''.']); end end
You can now solve the IVP without entering any arguments from the command line.
In this example the ode23
function looks to the vdp1
Mfile to supply the missing arguments. Note that, once you've called odeset
to define options
, the calling syntax
also works, and that any options supplied via the command line override corresponding options specified in the Mfile (see odeset
).
See Also
The MATLAB Version 5 help entries for the ODE solvers and their associated functions: ode23
, ode45
, ode113
, ode15s
, ode23s
, ode23t
, ode23tb
, odeget
, odeset
Type at the MATLAB command line: more on, type function, more off
. The Version 5 help follows the Version 6 help.
ode45, ode23, ode113, ode15s, ode23s, ode23t, ode23tb  odeget 