Explain-improved euler methods

Question

Use Euler and the Improved Euler methods.

1. Consider the initial value problem y' = 2xy, y(1) = 1. Use the Euler's method and improved Euler's method with h = 0.1and h = 0.05 to obtain approximate values of the solution at x = 1.5. At each step compare the approximate value with the actual value of the analytic solution. Also calculate the bound for thelocal truncation error using the formula:
yⁿ(c) h²/2, where x_n n+1

2. Consider the initial value problem ' ( 1) , (0) 2. 2 y = x + y - y = Use the improved Euler's method with h = 0.1and h = 0.05 to obtain approximate values of the solution at x = 0.5. At each step compare the approximate value with the actual value of the analytic solution.