'''simple docstring'''


from collections.abc import Callable

import numpy as np


def  _lowerCamelCase   (				lowerCamelCase_ :							Callable     ,					lowerCamelCase_ :							float     ,					lowerCamelCase_ :							float     ,					lowerCamelCase_ :							float     ,					lowerCamelCase_ :							float       ):



     """simple docstring"""





     UpperCAmelCase_							:       Optional[int]       =				int(np.ceil((x_end - xa) / step_size       )       )
     UpperCAmelCase_							:       int       =				np.zeros((n + 1,)       )
     UpperCAmelCase_							:       List[str]       =				ya
     UpperCAmelCase_							:       List[Any]       =				xa

     for k in range(lowerCamelCase_       ):
          UpperCAmelCase_							:       int       =				y[k] + step_size * ode_func(lowerCamelCase_     ,					y[k]       )
          UpperCAmelCase_							:       str       =				y[k] + (
              (step_size / 2) * (ode_func(lowerCamelCase_     ,					y[k]       ) + ode_func(x + step_size     ,					lowerCamelCase_       ))
          )
          x += step_size

     return y


if __name__ == "__main__":
 import doctest

 doctest.testmod()







'''simple docstring'''


def  _lowerCamelCase   (				lowerCamelCase_ :							str     ,					lowerCamelCase_ :							str = " "       ):



     """simple docstring"""





     UpperCAmelCase_							:       str       =				[]

     UpperCAmelCase_							:       List[Any]       =				0
     for index, char in enumerate(lowerCamelCase_       ):
          if char == separator:
               split_words.append(string[last_index:index]       )
               UpperCAmelCase_							:       Optional[Any]       =				index + 1
          elif index + 1 == len(lowerCamelCase_       ):
               split_words.append(string[last_index : index + 1]       )
     return split_words


if __name__ == "__main__":
 from doctest import testmod

 testmod()






