Solve the following ODE, y' + 2y = x+4, y(0) = 1 using. a. Approximate y (0.4) second order Runge-Kutta with h=0.2 and y(0)=0. b. Using the previous results use Adam-Bashforth(predictor) two step explicit method in conjunction with Adam-Moulton two step implicit method (corrector) to approximate y (0.8) with h=0.01 c. Use MATLAB to approximate and plot the solution from x=0 to x=1 using 2nd order Runge-Kutta with h=0.2 and y (0) =1

(Numerical analysis) Here’s a challenging problem for those who know a little calculus. The Newton-Raphson method can be used to find the roots of any equation y(x)=0. In this method, the (i+1)stapproximation,xi+1,toarootofy(x)=0 is given in terms of the ith approximation, xi, by the following formula, where y’ denotes the derivative of y(x) with respect to x: xi+1=xiy(xi)/y(xi) For example, if y(x)=3x2+2x2,theny(x)=6x+2 , and the roots are found by making a reasonable guess for a first approximation x1 and iterating by using this equation: xi+1=xi(3xi2+2xi2)/(6xi+2) a. Using the Newton-Raphson method, find the two roots of the equation 3x2+2x2=0. (Hint: There’s one positive root and one negative root.) b. Extend the program written for Exercise 6a so that it finds the roots of any function y(x)=0, when the function for y(x) and the derivative of y(x) are placed in the code.