Calculator to get the approximate solution at x 3 by

Question

Question 1: Consider the initial-value problem:

y'= 1 + (x -y)², 2 ≤ x ≤ 3 ; y(2) = 1

Its actual solution is y(x) = x + 1/(1 - x ). Use h = 1/2 and a calculator to get the approximate solution at x = 3 by applying

a. Euler's method
b. RK2
c. Modified Euler method
d. RK4

Then, compare their results with the actual value y(3) = 2.5.

Question 2: Consider the following system of first-order differential equations:

u₁' = u₂ - u₃ + t             u₁(0) = 1  

u₂' = 3t²                             u₂(0) = 1           0 ≤ t ≤1

u₃' = u₂ + e^-t                     u₃(0) = -1

The actual solution is

u₁(t) = -t⁵/20 + t⁴/4 + t + 2 - e^-t

u₂(t) = t³ + 1

u₃(t) = t⁴/4 + t - e^-t

Use "RK4SYSTEM" to approximate the solution with h = 0.2, 0.1, 0.05, and compare the errors to see if you can conclude that RK4SYSTEM is a fourth-order method for systems of differential equations.