### `tf.contrib.integrate.odeint(func, y0, t, rtol=1e-06, atol=1e-12, method=None, options=None, full_output=False, name=None)` {#odeint}

Integrate a system of ordinary differential equations.

Solves the initial value problem for a non-stiff system of first order ode-s:

  ```
  dy/dt = func(y, t), y(t[0]) = y0
  ```

where y is a Tensor of any shape.

For example:

  ```
  # solve `dy/dt = -y`, corresponding to exponential decay
  tf.contrib.integrate.odeint(lambda y, _: -y, 1.0, [0, 1, 2])
  => [1, exp(-1), exp(-2)]
  ```

Output dtypes and numerical precision are based on the dtypes of the inputs
`y0` and `t`.

Currently, implements 5th order Runge-Kutta with adaptive step size control
and dense output, using the Dormand-Prince method. Similar to the 'dopri5'
method of `scipy.integrate.ode` and MATLAB's `ode45`.

Based on: Shampine, Lawrence F. (1986), "Some Practical Runge-Kutta Formulas",
Mathematics of Computation, American Mathematical Society, 46 (173): 135-150,
doi:10.2307/2008219

##### Args:


*  <b>`func`</b>: Function that maps a Tensor holding the state `y` and a scalar Tensor
    `t` into a Tensor of state derivatives with respect to time.
*  <b>`y0`</b>: N-D Tensor giving starting value of `y` at time point `t[0]`. May
    have any floating point or complex dtype.
*  <b>`t`</b>: 1-D Tensor holding a sequence of time points for which to solve for
    `y`. The initial time point should be the first element of this sequence,
    and each time must be larger than the previous time. May have any floating
    point dtype. If not provided as a Tensor, converted to a Tensor with
    float64 dtype.
*  <b>`rtol`</b>: optional float64 Tensor specifying an upper bound on relative error,
    per element of `y`.
*  <b>`atol`</b>: optional float64 Tensor specifying an upper bound on absolute error,
    per element of `y`.
*  <b>`method`</b>: optional string indicating the integration method to use. Currently,
    the only valid option is `'dopri5'`.
*  <b>`options`</b>: optional dict of configuring options for the indicated integration
    method. Can only be provided if a `method` is explicitly set. For
    `'dopri5'`, valid options include:
    * first_step: an initial guess for the size of the first integration
      (current default: 1.0, but may later be changed to use heuristics based
      on the gradient).
    * safety: safety factor for adaptive step control, generally a constant
      in the range 0.8-1 (default: 0.9).
    * ifactor: maximum factor by which the adaptive step may be increased
      (default: 10.0).
    * dfactor: maximum factor by which the adpative step may be decreased
      (default: 0.2).
    * max_num_steps: integer maximum number of integrate steps between time
      points in `t` (default: 1000).
*  <b>`full_output`</b>: optional boolean. If True, `odeint` returns a tuple
    `(y, info_dict)` describing the integration process.
*  <b>`name`</b>: Optional name for this operation.

##### Returns:


*  <b>`y`</b>: (N+1)-D tensor, where the first dimension corresponds to different
    time points. Contains the solved value of y for each desired time point in
    `t`, with the initial value `y0` being the first element along the first
    dimension.
*  <b>`info_dict`</b>: only if `full_output == True`. A dict with the following values:
    * num_func_evals: integer Tensor counting the number of function
      evaluations.
    * integrate_points: 1D float64 Tensor with the upper bound of each
      integration time step.
    * error_ratio: 1D float Tensor with the estimated ratio of the integration
      error to the error tolerance at each integration step. An ratio greater
      than 1 corresponds to rejected steps.

##### Raises:


*  <b>`ValueError`</b>: if an invalid `method` is provided.
*  <b>`TypeError`</b>: if `options` is supplied without `method`, or if `t` or `y0` has
    an invalid dtype.

