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Solve moderately stiff ODEs and DAEs — trapezoidal rule

```
[t,y] =
ode23t(odefun,tspan,y0)
```

```
[t,y] =
ode23t(odefun,tspan,y0,options)
```

```
[t,y,te,ye,ie]
= ode23t(odefun,tspan,y0,options)
```

`sol = ode23t(___)`

`[`

,
where `t`

,`y`

] =
ode23t(`odefun`

,`tspan`

,`y0`

)`tspan = [t0 tf]`

, integrates the system of
differential equations $$y\text{'}=f\left(t,y\right)$$ from `t0`

to `tf`

with
initial conditions `y0`

. Each row in the solution
array `y`

corresponds to a value returned in column
vector `t`

.

All MATLAB^{®} ODE solvers can solve systems of equations of
the form $$y\text{'}=f\left(t,y\right)$$,
or problems that involve a mass matrix, $$M\left(t,y\right)y\text{'}=f\left(t,y\right)$$.
The solvers all use similar syntaxes. The `ode23s`

solver
only can solve problems with a mass matrix if the mass matrix is constant. `ode15s`

and `ode23t`

can
solve problems with a mass matrix that is singular, known as differential-algebraic
equations (DAEs). Specify the mass matrix using the `Mass`

option
of `odeset`

.

`[`

additionally
finds where functions of (`t`

,`y`

,`te`

,`ye`

,`ie`

]
= ode23t(`odefun`

,`tspan`

,`y0`

,`options`

)*t*,*y*),
called event functions, are zero. In the output, `te`

is
the time of the event, `ye`

is the solution at the
time of the event, and `ie`

is the index of the triggered
event.

For each event function, specify whether the integration is
to terminate at a zero and whether the direction of the zero crossing
matters. Do this by setting the `'Events'`

property
to a function, such as `myEventFcn`

or `@myEventFcn`

,
and creating a corresponding function: [`value`

,`isterminal`

,`direction`

]
= `myEventFcn`

(`t`

,`y`

).
For more information, see ODE Event Location.

returns
a structure that you can use with `sol`

= ode23t(___)`deval`

to evaluate
the solution at any point on the interval `[t0 tf]`

.
You can use any of the input argument combinations in previous syntaxes.

`ode23t`

is an implementation of the trapezoidal
rule using a “free” interpolant. This solver is preferred
over `ode15s`

if the problem is only moderately stiff
and you need a solution without numerical damping. `ode23t`

also
can solve differential algebraic equations (DAEs) [1], [2].

[1] Shampine, L. F., M. W. Reichelt, and J.A.
Kierzenka, “Solving
Index-1 DAEs in MATLAB and Simulink”, *SIAM
Review*, Vol. 41, 1999, pp. 538–552.

[2] Shampine, L. F. and M. E. Hosea, “Analysis
and Implementation of TR-BDF2,” *Applied Numerical
Mathematics 20*, 1996.

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