import numpy as np
import matplotlib.pyplot as plt
 
# Define the function f(x, y) that represents the ODE
def f(x, y):
    # Edit the equation here to represent dy/dx = f(x, y)
    return x**2 - y  # Example: dy/dx = x^2 - y
 
# Euler's method implementation
def euler_method(f, y0, x0, xf, h):
    # Number of steps
    num_steps = int((xf - x0) / h) + 1
    xs = np.linspace(x0, xf, num_steps)  # x values
    ys = np.zeros(num_steps)               # y values
    ys[0] = y0                               # Set initial condition
 
    # Iterative calculation using Euler's method
    for i in range(num_steps - 1):
        ys[i + 1] = ys[i] + h * f(xs[i], ys[i])
 
    return xs, ys
 
# Inputs: Initial conditions and range
x0 = 0                     # Start value of x
y0 = 1                     # Initial value of y (y(x0))
xf = 5                     # End value of x
h = 0.1                    # Step size
 
# Solve ODE using Euler's method
xs, ys = euler_method(f, y0, x0, xf, h)
 
# Display the solution as a list of points
print("x values:", xs)
print("y values:", ys)
 
# Plot the results
plt.plot(xs, ys, marker='o', linestyle='-', color='b', label="Approximate Solution (Euler's Method)")
plt.title("Euler's Method for Solving 1st Order ODE")
plt.xlabel("x")
plt.ylabel("y")
plt.grid(True)
plt.legend()
plt.show()

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