an is assumed to depend on both
          t and y; it will be called as ``jac(t, y)`` as necessary.
          For the 'Radau' and 'BDF' methods, the return value might be a
          sparse matrix.
        * If None (default), the Jacobian will be approximated by
          finite differences.

    It is generally recommended to provide the Jacobian rather than
    relying on a finite-difference approximation.
jac_sparsity : {None, array_like, sparse matrix}, optional
    Defines a sparsity structure of the Jacobian matrix for a
    finite-difference approximation. Its shape must be (n, n). This argument
    is ignored if `jac` is not `None`. If the Jacobian has only few non-zero
    elements in *each* row, providing the sparsity structure will greatly
    speed up the computations [2]_. A zero entry means that a corresponding
    element in the Jacobian is always zero. If None (default), the Jacobian
    is assumed to be dense.
vectorized : bool, optional
    Whether `fun` can be called in a vectorized fashion. Default is False.

    If ``vectorized`` is False, `fun` will always be called with ``y`` of
    shape ``(n,)``, where ``n = len(y0)``.

    If ``vectorized`` is True, `fun` may be called with ``y`` of shape
    ``(n, k)``, where ``k`` is an integer. In this case, `fun` must behave
    such that ``fun(t, y)[:, i] == fun(t, y[:, i])`` (i.e. each column of
    the returned array is the time derivative of the state corresponding
    with a column of ``y``).

    Setting ``vectorized=True`` allows for faster finite difference
    approximation of the Jacobian by this method, but may result in slower
    execution overall in some circumstances (e.g. small ``len(y0)``).

Attributes
----------
n : int
    Number of equations.
status : string
    Current status of the solver: 'running', 'finished' or 'failed'.
t_bound : float
    Boundary time.
direction : float
    Integration direction: +1 or -1.
t : float
    Current time.
y : ndarray
    Current state.
t_old : float
    Previous time. None if no steps were made yet.
step_size : float
    Size of the last successful step. None if no steps were made yet.
nfev : int
    Number of evaluations of the right-hand side.
njev : int
    Number of evaluations of the Jacobian.
nlu : int
    Number of LU decompositions.

References
----------
.. [1] E. Hairer, G. Wanner, "Solving Ordinary Differential Equations II:
       Stiff and Differential-Algebraic Problems", Sec. IV.8.
.. [2] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of
       sparse Jacobian matrices", Journal of the Institute of Mathematics
       and its Applications, 13, pp. 117-120, 1974.
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