Step 1 - Data and Initialization:

You have an ordinary differential equation (ODE) of the form: y'_n = f(x_n, y_n), and an initial condition: y₀ = y₀. You want to numerically solve the ODE.

Step 2 - Choose a Step Size:

Select a small step size, h, which determines the size of the discrete steps used to approximate the solution.

Step 3 - Iteration (Euler's Method):

Apply Euler's method to update the solution at each step:

y_n+1 = y_n + h * f(x_n, y_n)

Here, y_n is the current approximation of y(x_n), and y_n+1 is the updated approximation at x_n+1 = x_n + h.

Step 3 - Iteration (Modified Euler's Method):

Apply the modified Euler's method, which is more accurate than Euler's method. It involves two steps for each iteration:

Predict the value at x_n+1 using a forward Euler step:

y^~_n+1 = y_n + h * f(x_n, y_n)

Correct the prediction using the midpoint of the interval:

y_n+1 = y_n + (h/2) * [f(x_n, y_n) + f(x_n+1, y^~_n+1)]

Step 4 - Convergence Check:

Assess the accuracy and convergence of the method by comparing the results for different step sizes.

Step 5 - Output:

The final result is the approximate solution to the ODE over the specified range.

Euler's method and modified Euler's method are basic but important tools for solving ODEs numerically. Modified Euler's method is more accurate than the standard Euler method but still relatively simple to implement.