In Problems 1– 4, construct a table comparing the indicated values of y(x) using Euler’s method, the improved Euler’s method, and the RK4 method. Compute to four rounded decimal places. Use h = 0.1 and then use h = 0.05.

1. y′ = 2 ln xy,     y(1) = 2;
   y(1.1), y(1.2), y(1.3), y(1.4), y(1.5)
2. y′ = sin x² + cos y²,     y(0) = 0;
   y(0.1), y(0.2), y(0.3), y(0.4), y(0.5)
3. y′ = ,     y(0.5) = 0.5;
   y(0.6), y(0.7), y(0.8), y(0.9), y(1.0)
4. y′ = xy + y²,     y(1) = 1;
   y(1.1), y(1.2), y(1.3), y(1.4), y(1.5)
5. Use Euler’s method to approximate y(0.2), where y(x) is the solution of the initial-value problem y″ − (2x + 1)y = 1, y(0) = 3, y′(0) = 1. First use one step with h = 0.2, and then repeat the calculations using two steps with h = 0.1.
6. Use the Adams–Bashforth–Moulton method to approximate y(0.4), where y(x) is the solution of the initial-value problem y′ = 4x − 2y, y(0) = 2. Use h = 0.1 and the RK4 method to compute y₁, y₂, and y₃.
7. Use Euler’s method with h = 0.1 to approximate x(0.2) and y(0.2), where x(t), y(t) is the solution of the initial-value problem

   

8. Use the finite difference method with n = 10 to approximate the solution of the boundary-value problem y″ + 6.55(1 + x)y = 1, y(0) = 0, y(1) = 0.