Write this problem as an initial value problem for a system

Question

1. This question concerns the problem

y' = x/y, Y(0) = 3

which has solution y(x) = √x² + 9.

(a) Use Euler's method with steps h = 1, 1/2 and 1/4 to approximate y(1). Show your calculations.

(b) Use Euler's method with steps h = 10^-k for k = 1, 2, 3, 4 to solve the initial value problem over [0, 1]. Plot the error of y(1) versus h using a log-log plot. Can you identify the order of the method from the plot? Provide a print-out your MATLAB code and graph in your answer.

(c) Use Heun's method with 50 steps to obtain a plot of y versus x for 0 ≤ x ≤ 3.

2. Provide a print-out your MATLAB code in your answers.

(a) Apply the 2-step Adams-Bashforth to the initial value problem of Question 1 to com¬pute y(3). Use 20 steps and start your calculation using the modified Euler method.

(b) Repeat (a) but this time use the 2-step Adams-Bashforth method and then the 2-step Adams-Moulton formula as a corrector. Start the calculation using the classical Runge-Kutta method.

3. The angle θ of a pendulum satisfies the initial value problem

d²θ/dt² = -g/l sin θ, θ(0) = Π/4, θ'(0) = 0.

(a) Write this problem as an initial value problem for a system of first order differential equations.

(b) Assuming that g = 9.8 and l = 1, use the classical RK4 method to solve the system. Plot 0(t) for 0 ≤ t ≤ 6 and create a phase plane plot (i.e. a plot of 0' versus 0).

4. Solve the Bessel equation x²y" + xy' + (x²-4)y = 0 with the boundary conditions y(1) = 1 and y(10) = 0.5 using the shooting method.

Provide a print-out of your MATLAB code and the calculation of the initial slope that you compute using the secant method.