#### Euler Modified

R
A
```from collections.abc import Callable

import numpy as np

def euler_modified(
ode_func: Callable, y0: float, x0: float, step_size: float, x_end: float
) -> np.array:
"""
Calculate solution at each step to an ODE using Euler's Modified Method
The Euler Method is straightforward to implement, but can't give accurate solutions.
So, some changes were proposed to improve accuracy.

https://en.wikipedia.org/wiki/Euler_method

Arguments:
ode_func -- The ode as a function of x and y
y0 -- the initial value for y
x0 -- the initial value for x
stepsize -- the increment value for x
x_end -- the end value for x

>>> # the exact solution is math.exp(x)
>>> def f1(x, y):
...     return -2*x*(y**2)
>>> y = euler_modified(f1, 1.0, 0.0, 0.2, 1.0)
>>> y[-1]
0.503338255442106
>>> import math
>>> def f2(x, y):
...     return -2*y + (x**3)*math.exp(-2*x)
>>> y = euler_modified(f2, 1.0, 0.0, 0.1, 0.3)
>>> y[-1]
0.5525976431951775
"""
n = int(np.ceil((x_end - x0) / step_size))
y = np.zeros((n + 1,))
y[0] = y0
x = x0

for k in range(n):
y_get = y[k] + step_size * ode_func(x, y[k])
y[k + 1] = y[k] + (
(step_size / 2) * (ode_func(x, y[k]) + ode_func(x + step_size, y_get))
)
x += step_size

return y

if __name__ == "__main__":
import doctest

doctest.testmod()
```