# scipy.integrate.odeint

scipy.integrate.odeint(func, y0, t, args=(), Dfun=None, col_deriv=0, full_output=0, ml=None, mu=None, rtol=None, atol=None, tcrit=None, h0=0.0, hmax=0.0, hmin=0.0, ixpr=0, mxstep=0, mxhnil=0, mxordn=12, mxords=5, printmessg=0, tfirst=False)

  Integrate a system of ordinary differential equations.

  Note

  For new code, use `scipy.integrate.solve_ivp` to solve a differential equation.

  Solve a system of ordinary differential equations using lsoda from the FORTRAN library odepack.

  Solves the initial value problem for stiff or non-stiff systems of first order ode-s:

      dy/dt = func(y, t, ...)  [or func(t, y, ...)]

  where y can be a vector.

  Note

  By default, the required order of the first two arguments of func are in the opposite order of the arguments in the system definition function used by the `scipy.integrate.ode` class and the function `scipy.integrate.solve_ivp`. To use a function with the signature `func(t, y, ...)`, the argument tfirst must be set to `True`.

  Parameters:
    func callable(y, t, …) or callable(t, y, …)

      Computes the derivative of y at t. If the signature is `callable(t, y, ...)`, then the argument tfirst must be set `True`.

    y0 array

      Initial condition on y (can be a vector).

    t array

      A sequence of time points for which to solve for y. The initial value point should be the first element of this sequence. This sequence must be monotonically increasing or monotonically decreasing; repeated values are allowed.

    args tuple, optional

      Extra arguments to pass to function.

    Dfun callable(y, t, …) or callable(t, y, …)

      Gradient (Jacobian) of func. If the signature is `callable(t, y, ...)`, then the argument tfirst must be set `True`.

    col_deriv bool, optional

      True if Dfun defines derivatives down columns (faster), otherwise Dfun should define derivatives across rows.

    full_output bool, optional

      True if to return a dictionary of optional outputs as the second output

    printmessg bool, optional

      Whether to print the convergence message

    tfirst bool, optional

      If True, the first two arguments of func (and Dfun, if given) must `t, y` instead of the default `y, t`.

      New in version 1.1.0.

  Returns:
    y array, shape (len(t), len(y0))

      Array containing the value of y for each desired time in t, with the initial value y0 in the first row.

    infodict dict, only returned if full_output == True

      Dictionary containing additional output information

      key | meaning
      --- | ---
      ‘hu’ | vector of step sizes successfully used for each time step
      ‘tcur’ | vector with the value of t reached for each time step (will always be at least as large as the input times)
      ‘tolsf’ | vector of tolerance scale factors, greater than 1.0, computed when a request for too much accuracy was detected
      ‘tsw’ | value of t at the time of the last method switch (given for each time step)
      ‘nst’ | cumulative number of time steps
      ‘nfe’ | cumulative number of function evaluations for each time step
      ‘nje’ | cumulative number of jacobian evaluations for each time step
      ‘nqu’ | a vector of method orders for each successful step
      ‘imxer’ | index of the component of largest magnitude in the weighted local error vector (e / ewt) on an error return, -1 otherwise
      ‘lenrw’ | the length of the double work array required
      ‘leniw’ | the length of integer work array required
      ‘mused’ | a vector of method indicators for each successful time step: 1: adams (nonstiff), 2: bdf (stiff)

  Other Parameters:
    ml, mu int, optional

      If either of these are not None or non-negative, then the Jacobian is assumed to be banded. These give the number of lower and upper non-zero diagonals in this banded matrix. For the banded case, Dfun should return a matrix whose rows contain the non-zero bands (starting with the lowest diagonal). Thus, the return matrix jac from Dfun should have shape `(ml + mu + 1, len(y0))` when `ml >=0` or `mu >=0`. The data in jac must be stored such that `jac[i - j + mu, j]` holds the derivative of the i`th equation with respect to the `j`th state variable. If `col_deriv is True, the transpose of this jac must be returned.

    rtol, atol float, optional

      The input parameters rtol and atol determine the error control performed by the solver. The solver will control the vector, e, of estimated local errors in y, according to an inequality of the form `max-norm of (e / ewt) <= 1`, where ewt is a vector of positive error weights computed as `ewt = rtol * abs(y) + atol`. rtol and atol can be either vectors the same length as y or scalars. Defaults to 1.49012e-8.

    tcrit ndarray, optional

      Vector of critical points (e.g., singularities) where integration care should be taken.

    h0 float, (0: solver-determined), optional

      The step size to be attempted on the first step.

    hmax float, (0: solver-determined), optional

      The maximum absolute step size allowed.

    hmin float, (0: solver-determined), optional

      The minimum absolute step size allowed.

    ixpr bool, optional

      Whether to generate extra printing at method switches.

    mxstep int, (0: solver-determined), optional

      Maximum number of (internally defined) steps allowed for each integration point in t.

    mxhnil int, (0: solver-determined), optional

      Maximum number of messages printed.

    mxordn int, (0: solver-determined), optional

      Maximum order to be allowed for the non-stiff (Adams) method.

    mxords int, (0: solver-determined), optional

      Maximum order to be allowed for the stiff (BDF) method.

  See also

  `solve_ivp`

    solve an initial value problem for a system of ODEs

  `ode`

    a more object-oriented integrator based on VODE

  `quad`

    for finding the area under a curve

  Examples

  The second order differential equation for the angle theta of a pendulum acted on by gravity with friction can be written:

      theta''(t) + b*theta'(t) + c*sin(theta(t)) = 0

  where b and c are positive constants, and a prime (’) denotes a derivative. To solve this equation with `odeint`, we must first convert it to a system of first order equations. By defining the angular velocity `omega(t) = theta'(t)`, we obtain the system:

      theta'(t) = omega(t)
      omega'(t) = -b*omega(t) - c*sin(theta(t))

  Let y be the vector [theta, omega]. We implement this system in Python as:

      >>> import numpy as np
      >>> def pend(y, t, b, c):
      ...     theta, omega = y
      ...     dydt = [omega, -b*omega - c*np.sin(theta)]
      ...     return dydt
      ...

  We assume the constants are b = 0.25 and c = 5.0:

      >>> b = 0.25
      >>> c = 5.0

  For initial conditions, we assume the pendulum is nearly vertical with theta(0) = pi - 0.1, and is initially at rest, so omega(0) = 0. Then the vector of initial conditions is

      >>> y0 = [np.pi - 0.1, 0.0]

  We will generate a solution at 101 evenly spaced samples in the interval 0 <= t <= 10. So our array of times is:

      >>> t = np.linspace(0, 10, 101)

  Call `odeint` to generate the solution. To pass the parameters b and c to pend, we give them to `odeint` using the args argument.

      >>> from scipy.integrate import odeint
      >>> sol = odeint(pend, y0, t, args=(b, c))

  The solution is an array with shape (101, 2). The first column is theta(t), and the second is omega(t). The following code plots both components.

      >>> import matplotlib.pyplot as plt
      >>> plt.plot(t, sol[:, 0], 'b', label='theta(t)')
      >>> plt.plot(t, sol[:, 1], 'g', label='omega(t)')
      >>> plt.legend(loc='best')
      >>> plt.xlabel('t')
      >>> plt.grid()
      >>> plt.show()