May someone help?

8. (15%) (Numerical Method for ODEs) Consider the initial value problem, dx dt = f(t, a) = - sin?(t)ac3 (t), t20, x(0) =1. (5) (a) (4%) Write the formula for the Euler method? Do one step of the Euler method with h = 0.1 to approximate x(0.1). (b) (6%) The second-order two-stage Runge-Kutta method is of the form K1 = f(th, k), K 2 = f (th + h, xk + hK1) Xk+1 = * * + h ( K 1 + , K2), In the above Runge-Kutta scheme, h > 0 is the time step, th = kh, k = 0, 1,2, ..., and the numerical solution xx approximates the exact solution x(t) for all k. Do one step of the second-order Runge-Kutta method with h = 0.1 to approximate x(0.1). P.4 of 5 (c) (5%) If one uses integration combined with the trapezoidal rule to ap- proximate (5), one gets Ck+1 - CK = 2 (f (th, Xx ) + f ( th+ 1 , XX+1 ) ). (6) Write down the recursive formula for the Newton's method in solving the nonlinear equation (6), in order to get Ck+1