An *ordinary differential equation* (ODE)
contains one or more derivatives of a dependent variable, *y*,
with respect to a single independent variable, *t*,
usually referred to as time. The notation used here for representing
derivatives of *y* with respect to *t* is $$y\text{'}$$ for
a first derivative, $$y\text{'}\text{'}$$ for
a second derivative, and so on. The *order* of
the ODE is equal to the highest-order derivative of *y* that
appears in the equation.

For example, this is a second order ODE:

$$y\text{'}\text{'}=9y$$

In an *initial value problem*, the ODE is
solved by starting from an initial state. Using the initial condition, $${y}_{0}$$,
as well as a period of time over which the answer is to be obtained, $$\left({t}_{0},{t}_{f}\right)$$,
the solution is obtained iteratively. At each step the solver applies
a particular algorithm to the results of previous steps. At the first
such step, the initial condition provides the necessary information
that allows the integration to proceed. The final result is that the
ODE solver returns a vector of time steps $$t=\left[{t}_{0},{t}_{1},{t}_{2},\mathrm{...},{t}_{f}\right]$$ as
well as the corresponding solution at each step $$y=\left[{y}_{0},{y}_{1},{y}_{2},\mathrm{...},{y}_{f}\right]$$.

The ODE solvers in MATLAB^{®} solve these types of first-order
ODEs:

Explicit ODEs of the form $$y\text{'}=f\left(t,y\right)$$.

Linearly implicit ODEs of the form $$M\left(t,y\right)y\text{'}=f\left(t,y\right)$$, where $$M\left(t,y\right)$$ is a nonsingular mass matrix. The mass matrix can be time- or state-dependent, or it can be a constant matrix. Linearly implicit ODEs involve linear combinations of the first derivative of

*y*, which are encoded in the mass matrix.Linearly implicit ODEs can always be transformed to an explicit form, $$y\text{'}={M}^{-1}\left(t,y\right)f\left(t,y\right)$$. However, specifying the mass matrix directly to the ODE solver avoids this transformation, which is inconvenient and can be computationally expensive.

If some components of $$y\text{'}$$ are missing, then the equations are called

*differential algebraic equations*, or DAEs, and the system of DAEs contains some*algebraic variables*. Algebraic variables are dependent variables whose derivatives do not appear in the equations. A system of DAEs can be rewritten as an equivalent system of first-order ODEs by taking derivatives of the equations to eliminate the algebraic variables. The number of derivatives needed to rewrite a DAE as an ODE is called the differential index. The`ode15s`

and`ode23t`

solvers can solve index-1 DAEs.Fully implicit ODEs of the form $$f\left(t,y,y\text{'}\right)=0$$. Fully implicit ODEs cannot be rewritten in an explicit form, and might also contain some algebraic variables. The

`ode15i`

solver is designed for fully implicit problems, including index-1 DAEs.

You can supply additional information to the solver for some
types of problems by using the `odeset`

function
to create an options structure.

You can specify any number of coupled ODE equations to solve,
and in principle the number of equations is only limited by available
computer memory. If the system of equations has *n* equations,

$$\left(\begin{array}{c}y{\text{'}}_{1}\\ y{\text{'}}_{2}\\ \vdots \\ y{\text{'}}_{n}\end{array}\right)=\left(\begin{array}{c}{f}_{1}\left(t,{y}_{1},{y}_{2},\mathrm{...},{y}_{n}\right)\\ {f}_{2}\left(t,{y}_{1},{y}_{2},\mathrm{...},{y}_{n}\right)\\ \vdots \\ {f}_{n}\left(t,{y}_{1},{y}_{2},\mathrm{...},{y}_{n}\right)\end{array}\right),$$

then the function that encodes the equations returns a vector
with *n* elements, corresponding to the values for $$y{\text{'}}_{1},\text{\hspace{0.17em}}y{\text{'}}_{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\dots \text{\hspace{0.17em}},\text{\hspace{0.17em}}y{\text{'}}_{n}$$.
For example, consider the system of two equations

$$\{\begin{array}{l}y{\text{'}}_{1}={y}_{2}\\ y{\text{'}}_{2}={y}_{1}\text{\hspace{0.17em}}{y}_{2}-2\text{\hspace{0.17em}}.\end{array}$$

A function that encodes these equations is

```
function dy = myODE(t,y)
dy(1) = y(2);
dy(2) = y(1)*y(2)-2;
```

The MATLAB ODE solvers only solve first-order equations. You must rewrite higher-order ODEs as an equivalent system of first-order equations using the generic substitutions

$$\begin{array}{l}{y}_{1}=y\\ {y}_{2}=y\text{'}\\ {y}_{3}=y\text{'}\text{'}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\vdots \\ {y}_{n}={y}^{(n-1)}.\end{array}$$

The result of these substitutions is a system of *n* first-order
equations

$$\{\begin{array}{l}y{\text{'}}_{1}={y}_{2}\\ y{\text{'}}_{2}={y}_{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\vdots \\ y{\text{'}}_{n}=f\left(t,{y}_{1},{y}_{2},\mathrm{...},{y}_{n}\right).\end{array}$$

For example, consider the third-order ODE

$$y\text{'}\text{'}\text{'}-y\text{'}\text{'}y+1=0.$$

Using the substitutions

$$\begin{array}{l}{y}_{1}=y\\ {y}_{2}=y\text{'}\\ {y}_{3}=y\text{'}\text{'}\end{array}$$

results in the equivalent first-order system

$$\{\begin{array}{l}y{\text{'}}_{1}={y}_{2}\\ y{\text{'}}_{2}={y}_{3}\\ y{\text{'}}_{3}={y}_{1}\text{\hspace{0.17em}}{y}_{3}-1.\end{array}$$

The code for this system of equations is then

```
function dydt = f(t,y)
dydt(1) = y(2);
dydt(2) = y(3);
dydt(3) = y(1)*y(3)-1;
```

Consider the complex ODE equation

$$y\text{'}=f\left(t,y\right)\text{\hspace{0.17em}},$$

where $$y={y}_{1}+i{y}_{2}$$. To solve it, separate the real and imaginary parts into different solution components, then recombine the results at the end. Conceptually, this looks like

$$\begin{array}{l}yv=\left[\text{Real}\left(y\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Imag}\left(y\right)\right]\\ fv=\left[\text{Real}\left(f\left(t,y\right)\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Imag}\left(f\left(t,y\right)\right)\right]\text{\hspace{0.17em}}.\end{array}$$

For example, if the ODE is $$y\text{'}=yt+2i$$, then you can represent the equation using a function file.

function f = complexf(t,y) % Define function that takes and returns complex values f = y.*t + 2*i;

Then, the code to separate the real and imaginary parts is

function fv = imaginaryODE(t,yv) % Construct y from the real and imaginary components y = yv(1) + i*yv(2); % Evaluate the function yp = complexf(t,y); % Return real and imaginary in separate components fv = [real(yp); imag(yp)];

When you run a solver to obtain the solution, the initial condition `y0`

is
also separated into real and imaginary parts to provide an initial
condition for each solution component.

y0 = 1+i; yv0 = [real(y0); imag(y0)]; tspan = [0 2]; [t,yv] = ode45(@imaginaryODE, tspan, yv0);

Once you obtain the solution, combine the real and imaginary components together to obtain the final result.

y = yv(:,1) + i*yv(:,2);

`ode45`

performs well with most ODE problems
and should generally be your first choice of solver. However, `ode23`

and `ode113`

can
be more efficient than `ode45`

for problems with
looser or tighter accuracy requirements.

Some ODE problems exhibit *stiffness*, or
difficulty in evaluation. Stiffness is a term that defies a precise
definition, but in general, stiffness occurs when there is a difference
in scaling somewhere in the problem. For example, if an ODE has two
solution components that vary on drastically different time scales,
then the equation might be stiff. You can identify a problem as stiff
if nonstiff solvers (such as `ode45`

) are unable
to solve the problem or are extremely slow. If you observe that a
nonstiff solver is very slow, try using a stiff solver such as `ode15s`

instead.
When using a stiff solver, you can improve reliability and efficiency
by supplying the Jacobian matrix or its sparsity pattern.

This table provides general guidelines on when to use each of the different solvers.

Solver | Problem Type | Accuracy | When to Use |
---|---|---|---|

`ode45` | Nonstiff | Medium | Most of the time. |

`ode23` | Low |
| |

`ode113` | Low to High |
| |

`ode15s` | Stiff | Low to Medium | Try |

`ode23s` | Low |
If there is a mass matrix, it must be constant. | |

`ode23t` | Low | Use
| |

`ode23tb` | Low | Like | |

`ode15i` | Fully implicit | Low | Use |

For details and further recommendations about when to use each solver, see [5].

There are several example files available that serve as excellent
starting points for most ODE problems. To run the **Differential
Equations Examples** app, which lets you easily explore and
run examples, type

odeexamples

To open an individual example file for editing, type

`edit exampleFileName.m`

To run an example, type

exampleFileName

This table contains a list of the available ODE and DAE example files as well as the solvers and options they use. Links are included for the subset of examples that are also published directly in the documentation.

Example File | Solver Used | Options Specified | Description | Documentation Link |
---|---|---|---|---|

`amp1dae` | `ode23t` | `'Mass'`
| Stiff DAE — electrical circuit with constant, singular mass matrix | Solve Stiff Differential Algebraic Equation |

`ballode` | `ode23` | `'Events'` `'OutputFcn'` `'OutputSel'` `'Refine'` `'InitialStep'` `'MaxStep'`
| Simple event location — bouncing ball | ODE Event Location |

`batonode` | `ode45` | `'Mass'`
| ODE with time- and state-dependent mass matrix — motion of a baton | — |

`brussode` | `ode15s` | `'JPattern'` `'Vectorized'`
| Stiff large problem — diffusion in a chemical reaction (the Brusselator) | Solve Stiff ODEs |

`burgersode` | `ode15s` | `'Mass'` `'MStateDependence'` `'JPattern'` `'MvPattern'` `'RelTol'` `'AbsTol'`
| ODE with strongly state-dependent mass matrix — Burgers' equation solved using a moving mesh technique | — |

`fem1ode` | `ode15s` | `'Mass'` `'MStateDependence'` `'Jacobian'`
| Stiff problem with a time-dependent mass matrix — finite element method | — |

`fem2ode` | `ode23s` | `'Mass'`
| Stiff problem with a constant mass matrix — finite element method | — |

`hb1ode` | `ode15s` | — | Stiff ODE problem solved on a very long interval — Robertson chemical reaction | — |

`hb1dae` | `ode15s` | `'Mass'` `'RelTol'` `'AbsTol'` `'Vectorized'`
| Stiff, linearly implicit DAE from a conservation law — Robertson chemical reaction | Solve Robertson Problem as Semi-Explicit Differential Algebraic Equations (DAEs) |

`ihb1dae` | `ode15i` | `'RelTol'` `'AbsTol'` `'Jacobian'`
| Stiff, fully implicit DAE — Robertson chemical reaction | Solve Robertson Problem as Implicit Differential Algebraic Equations (DAEs) |

`iburgersode` | `ode15i` | `'RelTol'` `'AbsTol'` `'Jacobian'` `'JPattern'`
| Implicit ODE system — Burgers' equation | — |

`kneeode` | `ode15s` | `'NonNegative'`
| The "knee problem" with nonnegativity constraints | Nonnegative ODE Solution |

`orbitode` | `ode45` | `'RelTol'` `'AbsTol'` `'Events'` `'OutputFcn'`
| Advanced event location — restricted three body problem | ODE Event Location |

`rigidode` | `ode45` | — | Nonstiff problem — Euler equations of a rigid body without external forces | Solve Nonstiff ODEs |

`vdpode` | `ode15s` | `'Jacobian'`
| Parameterizable van der Pol equation (stiff for large | Solve Stiff ODEs |

[1] Shampine, L. F. and M. K. Gordon, *Computer
Solution of Ordinary Differential Equations: the Initial Value Problem*,
W. H. Freeman, San Francisco, 1975.

[2] Forsythe, G., M. Malcolm, and C. Moler, *Computer
Methods for Mathematical Computations*, Prentice-Hall,
New Jersey, 1977.

[3] Kahaner, D., C. Moler, and S. Nash, *Numerical
Methods and Software*, Prentice-Hall, New Jersey, 1989.

[4] Shampine, L. F., *Numerical Solution
of Ordinary Differential Equations*, Chapman & Hall,
New York, 1994.

[5] Shampine, L. F. and M. W. Reichelt, "The
MATLAB ODE Suite," *SIAM Journal on Scientific Computing*,
Vol. 18, 1997, pp. 1–22.

[6] Shampine, L. F., Gladwell, I. and S. Thompson, *Solving
ODEs with MATLAB*, Cambridge University Press, Cambridge
UK, 2003.

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