



def  lowerCamelCase_	(      lowerCAmelCase__	:	list[int]			,	lowerCAmelCase__	:	list[int]				)       ->							tuple[float, float]:

							'''simple docstring'''






							if not len(lowerCAmelCase__				) == len(lowerCAmelCase__				) == 3:
														raise ValueError('Please enter a valid equation.'				)
							if equationa[0] == equationa[1] == equationa[0] == equationa[1] == 0:
														raise ValueError('Both a & b of two equations can\'t be zero.'				)

							# Extract the coefficients
							A						,   A						,   A      						=    equationa
							A						,   A						,   A      						=    equationa

							# Calculate the determinants of the matrices
							A      						=    aa * ba - aa * ba
							A      						=    ca * ba - ca * ba
							A      						=    aa * ca - aa * ca

							# Check if the system of linear equations has a solution (using Cramer's rule)
							if determinant == 0:
														if determinant_x == determinant_y == 0:
																					raise ValueError('Infinite solutions. (Consistent system)'				)
														else:
																					raise ValueError('No solution. (Inconsistent system)'				)
							else:
														if determinant_x == determinant_y == 0:
																					# Trivial solution (Inconsistent system)
																					return (0.0, 0.0)
														else:
																					A      						=    determinant_x / determinant
																					A      						=    determinant_y / determinant
																					# Non-Trivial Solution (Consistent system)
																					return (x, y)




from collections.abc import Callable

import numpy as np






def  lowerCamelCase_	(      lowerCAmelCase__	:	Callable			,	lowerCAmelCase__	:	float			,	lowerCAmelCase__	:	float			,	lowerCAmelCase__	:	float			,	lowerCAmelCase__	:	float				)       ->							np.array:

							'''simple docstring'''






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

							for k in range(lowerCAmelCase__				):
														A      						=    y[k] + step_size * ode_func(lowerCAmelCase__			,	y[k]				)
														A      						=    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()