`ode45`

Default Runge-Kutta solver for nonstiff ordinary differential equations.

Category: Mathematics > Numerical Integration and Differential Equations

Syntax

[t, y] = ode45(odefun, tspan, y0)

Integrates y' = odefun(t, y) over tspan from y0.

[t, y] = ode45(odefun, tspan, y0, options)

Also applies options from odeset (tolerances, events, output functions).

sol = ode45(___)

Returns a structure for use with deval.

[t, y, te, ye, ie] = ode45(___)

Also returns event times, states and indices when an event function is set.

Description

ode45 uses the Dormand-Prince pair of fourth- and fifth-order formulas: the difference between the two estimates controls the step size. It is a good first attempt for most problems, as long as they are not stiff.

Higher-order equations are written as first-order systems: for x'' = -x, use y = [x; x'] and odefun = @(t,y) [y(2); -y(1)]. If the integration needs huge numbers of tiny steps, the problem is probably stiff and ode15s is the right tool.

Input arguments

odefun: Handle dydt = odefun(t, y) returning a column vector the size of y.
tspan: [t0 tf], or a vector of increasing (or decreasing) times to report.
y0: Vector of initial values.
options: Structure from odeset. Default tolerances are RelTol = 1e-3 and AbsTol = 1e-6.

Output arguments

t: Column vector of time points.
y: Array with one row per time and one column per component.

Example

Harmonic oscillator

[t, y] = ode45(@(t, y) [y(2); -y(1)], [0 pi/2], [0; 1]);
fprintf('%.3f %.3f\n', y(end, 1), y(end, 2))

Output

1.000 -0.000

ode45