code: def runge_kutta_step(func, y0, f0, t0, dt):
      # Dopri5 Butcher tableaux
      alpha = jnp.array(
          [1 / 5, 3 / 10, 4 / 5, 8 / 9, 1.0, 1.0, 0], dtype=dt.dtype
      )
      beta = jnp.array(
          [
              [1 / 5, 0, 0, 0, 0, 0, 0],
              [3 / 40, 9 / 40, 0, 0, 0, 0, 0],
              [44 / 45, -56 / 15, 32 / 9, 0, 0, 0, 0],
              [
                  19372 / 6561,
                  -25360 / 2187,
                  64448 / 6561,
                  -212 / 729,
                  0,
                  0,
                  0,
              ],
              [
                  9017 / 3168,
                  -355 / 33,
                  46732 / 5247,
                  49 / 176,
                  -5103 / 18656,
                  0,
                  0,
              ],
              [
                  35 / 384,
                  0,
                  500 / 1113,
                  125 / 192,
                  -2187 / 6784,
                  11 / 84,
                  0,
              ],
          ],
          dtype=f0.dtype,
      )
      c_sol = jnp.array(
          [
              35 / 384,
              0,
              500 / 1113,
              125 / 192,
              -2187 / 6784,
              11 / 84,
              0,
          ],
          dtype=f0.dtype,
      )
      c_error = jnp.array(
          [
              35 / 384 - 1951 / 21600,
              0,
              500 / 1113 - 22642 / 50085,
              125 / 192 - 451 / 720,
              -2187 / 6784 - -12231 / 42400,
              11 / 84 - 649 / 6300,
              -1.0 / 60.0,
          ],
          dtype=f0.dtype,
      )
  
      def body_fun(i, k):
          ti = t0 + dt * alpha[i - 1]
          yi = y0 + dt.astype(f0.dtype) * jnp.dot(beta[i - 1, :], k)
          ft = func(yi, ti)
          return k.at[i, :].set(ft)
  
      k = jnp.zeros((7, f0.shape[0]), f0.dtype).at[0, :].set(f0)
      k = lax.fori_loop(1, 7, body_fun, k)
  
      y1 = dt.astype(f0.dtype) * jnp.dot(c_sol, k) + y0
      y1_error = dt.astype(f0.dtype) * jnp.dot(c_error, k)
      f1 = k[-1]
      return y1, f1, y1_error, k
  
imports: from functools import partial
  import operator
  import jax
  import jax.numpy
  from jax._src import core
  from jax import custom_derivatives
  from jax import lax
  from jax._src.numpy.util import promote_dtypes_inexact
  from jax._src.util import safe_map, safe_zip
  from jax.flatten_util import ravel_pytree
  from jax.tree_util import tree_leaves, tree_map
  from jax._src import linear_util
label_desc: Use image node_id 6 for calling a global function with example usage: runge_kutta_step(func, y0, f0, t0, dt) and returns: y1,  f1,  y1_error,  k, k