
"""simple docstring"""





def  lowerCamelCase__						(     __snake_case      )				->						str:


					"""simple docstring"""





					_UpperCamelCase									=    int(__snake_case      )
					if decimal in (0, 1):  # Exit cases for the recursion
										return str(__snake_case      )
					_UpperCamelCase					,    _UpperCamelCase									=    divmod(__snake_case,	2      )
					return binary_recursive(__snake_case      ) + str(__snake_case      )



def  lowerCamelCase__						(     __snake_case      )				->						str:


					"""simple docstring"""





					_UpperCamelCase									=    str(__snake_case      ).strip()
					if not number:
										raise ValueError('''No input value was provided'''      )
					_UpperCamelCase									=    '''-''' if number.startswith('''-'''      ) else ''''''
					_UpperCamelCase									=    number.lstrip('''-'''      )
					if not number.isnumeric():
										raise ValueError('''Input value is not an integer'''      )
					return F'''{negative}0b{binary_recursive(int(__snake_case      )      )}'''


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

				testmod()








"""simple docstring"""





from collections.abc import Callable

import numpy as np



def  lowerCamelCase__						(     __snake_case,	__snake_case,	__snake_case,	__snake_case,	__snake_case      )				->						np.array:


					"""simple docstring"""





					_UpperCamelCase									=    int(np.ceil((x_end - xa) / step_size      )      )
					_UpperCamelCase									=    np.zeros((n + 1,)      )
					_UpperCamelCase									=    ya
					_UpperCamelCase									=    xa

					for k in range(__snake_case      ):
										_UpperCamelCase									=    y[k] + step_size * ode_func(__snake_case,	y[k]      )
										_UpperCamelCase									=    y[k] + (
										    (step_size / 2) * (ode_func(__snake_case,	y[k]      ) + ode_func(x + step_size,	__snake_case      ))
										)
										x += step_size

					return y


if __name__ == "__main__":
				import doctest

				doctest.testmod()






