Runge-Kutta\tMethods\tfor\tmultiple\tequations The\t4th\torder\tRunge-Kutta\talgorithm\tis\tone\tof\tthe\tmore\tpopular\tself-starting numerical\talgorithms\tfor\tintegrating\tIVP's. For\ta\tset\tof\t1st\torder\tODE's\tit\tcan\tbe written\tas 1 1 1 1 yi+1 = yi + k1 + k2 + k3 + k4 h 3 3 6 6 = yi + 1 k + 2k2 + 2k3 + k4 h 6 1 ( ) where k1 = f (ti , yi ) 1 1 k2 = f ti + h, yi + k1h 2 2 1 1 k3 = f ti + h, yi + k2 h 2 2 ( k4 = f ti + h, yi + k3h ) where\tthe\tbarred\tterms yi , k1 , k2 , k3 , k4 become\tvectors\tof\tlength\tcorresponding\tto the\tnumber\tof\tDE's. For\tthe\tfollowing\tIVP\tproblem: d2y 2 2 + y = ( x 1); dx y(0) = 1, y '(0) = 0 i. Transpose\tthe\tproblem\tto\ttwo\t1st\torder\tequations. Write\tout\tthe\talgorithm\tas\t2-dimensional\tvectors. ii. Carry\tout\tone\tintegration\tstep. Use\ta\tstep\tsize\th\t=\t0.2