can you help me write python code for euler's method of approximation?  
I would like to be able to input an expression for the differential equation, 
an input for the x and y coordinates of the initial condition, an input for the ending x-coordinate, 
and an input for the step size.


can you write this code without any imports.  I'd like it to run by itself.  
I am using python playground to test the code https://programiz.pro/ide/python


I don't know why the math function is not working in python playground, is there a workaround for this?


prior to running the code for the approximations, I would like user prompts for each of the following:   
1.  an expression for the differential equation, 2.  prompts for the initial conditions coordinates...x and y,  
3.  a prompt for the ending x-value, and 4. a prompt for the step size.

I'd like to continue improving this code...namely is the differential equations expression.  The code seems to work fine 
when the expression is in terms of x, but I have experienced an error when the expression is expressed in terms of y.  I am 
using the differential expression 3y-1 with an initial x-value of 0 and an initial y-value of 1.  When estimating the 
value when x = 1 using a step size of .25, I expect a value of 6.5859375.  Can you help me adjust the code so that this happens.

I ran the code again, and here are the error messages:


Copy
File "<exec>", line 62, in <module>
  File "<exec>", line 43, in euler_method
  File "<exec>", line 16, in eval_expression
    3y-1
    ^
SyntaxError: invalid decimal literal
*********************************************************************************************************************


import math
import re

def sin(x):
    return math.sin(x)

def cos(x):
    return math.cos(x)

def exp(x):
    return math.exp(x)

def prepare_expression(expr):
    """Prepare the expression by adding explicit multiplication operators."""
    # Add * for implied multiplication like 3y -> 3*y
    expr = re.sub(r'(\d+)([xy])', r'\1*\2', expr)
    # Add * for implied multiplication like yx -> y*x
    expr = re.sub(r'([xy])([xy])', r'\1*\2', expr)
    return expr

def eval_expression(expr, x, y):
    """Evaluate the expression string with given x and y values."""
    prepared_expr = prepare_expression(expr)
    # Create a local namespace with x, y, and math functions
    local_namespace = {"x": x, "y": y, "sin": sin, "cos": cos, "exp": exp}
    return eval(prepared_expr, {}, local_namespace)

def euler_method(f_str, x0, y0, x_end, h):
    """
    Implement Euler's method of approximation.
    
    Parameters:
    f_str (str): String representation of the differential equation dy/dx = f(x, y)
    x0 (float): Initial x-coordinate
    y0 (float): Initial y-coordinate
    x_end (float): Ending x-coordinate
    h (float): Step size
    
    Returns:
    tuple: Lists of x and y values
    """
    # Calculate the number of steps
    n = int((x_end - x0) / h)
    
    # Initialize lists to store x and y values
    x_values = [x0]
    y_values = [y0]
    
    # Current values
    x_current = x0
    y_current = y0
    
    # Perform Euler's method
    for i in range(n):
        # Calculate slope at current point
        slope = eval_expression(f_str, x_current, y_current)
        
        # Calculate next values
        x_next = x_current + h
        y_next = y_current + h * slope
        
        # Store the new values
        x_values.append(x_next)
        y_values.append(y_next)
        
        # Update current values for next iteration
        x_current = x_next
        y_current = y_next
    
    return x_values, y_values

# User prompts
print("Welcome to the Euler's Method Approximation Program")
print("Please enter the following information:")

f_str = input("1. Enter the expression for the differential equation (dy/dx = f(x,y)): ")
x0 = float(input("2. Enter the initial x-coordinate: "))
y0 = float(input("3. Enter the initial y-coordinate: "))
x_end = float(input("4. Enter the ending x-value: "))
h = float(input("5. Enter the step size: "))

# Run Euler's method
x_values, y_values = euler_method(f_str, x0, y0, x_end, h)

# Print results
print("\nResults:")
for x, y in zip(x_values, y_values):
    print(f"x = {x:.2f}, y = {y:.6f}")

print(f"\nFinal approximation at x = {x_end}: y = {y_values[-1]:.7f}")