6.5 A Nonline ar Differential Equation for an Orbit of a Satelli Consider the problem of an orbit of a satellite, whose position and velocity are obtained as the solution of the following state where G - 6.672 x 10-I N m/kg is the gravitational constant, and Me 5.97 x 104 kg is the mass of the earth. Note that (xx2) and x, ) denote the position and velocity, respectively, of the satellite on the pland having the earth at its origin. This state equation is defined in the M-file dr sat.m below (a) Supplement the following program "nnpo5. which uses the three routines ode RK4(). ode45(), and ode23) to find the poths of the satellite with the following initial positions/velocities for one day function dx-at-sat ( tx) if r c" Ro, return ; ond% when colliding bnm6pos.n to solve a nonlinear d.e. on the orbit of a satellits global G Ne Re to 0; T 24*60-60: t T;N 2000 V20s (3071 3500 2000]: for iter t:length(v20s) x0 [x10 x20 v10 v201; tol. 1e-6; tR,xR) ode RK4tf, (t0 tri.xo,N: [t23s, x2351 ode23s (7777 (i) (x10?X1o) = (4.223 102.0)[m] and (x30. x40) s(h-no) = (0. 307 1)[m/s) (0. 3500)Im/s]. (0. 2000)[m/s]. (ii) (x10,X20) = (4.223 10. 0)[m] and (x30. xao) = (ulo-vo) = Run the program and check if the plotting results are as depicted in Fig. P6.5 (b) In Fig. P6.5, we see that the "ode23s()" solution path differs from the others for case (a) and the "ode45)" and "ode23s()" paths differ from the "ode_RK4()" path for case (ii). But, we do not know which one is more accurate. In order to find which one is the closest to the true solution, apply the two routines "ode45()" and "ode23s(" with smaller relative error tolerance of tol 10-6 to find the paths for the three cases. Which one do you think is the closest to the true solution among the paths obtained in (a)? but to warn the users of the danger of abusing them. With smaller number of steps (N) (ie.. larger step size), the routine "ode RK4 (0" will also deviate much from the true sulution. The MATLAB buili-in ruutines have too many good features to be mentioned here. Note that setting the parameters such as (cf) The purpose of this problem is not to compure the several MATLAB routines 6.5 A Nonline ar Differential Equation for an Orbit of a Satelli Consider the problem of an orbit of a satellite, whose position and velocity are obtained as the solution of the following state where G - 6.672 x 10-I N m/kg is the gravitational constant, and Me 5.97 x 104 kg is the mass of the earth. Note that (xx2) and x, ) denote the position and velocity, respectively, of the satellite on the pland having the earth at its origin. This state equation is defined in the M-file dr sat.m below (a) Supplement the following program "nnpo5. which uses the three routines ode RK4(). ode45(), and ode23) to find the poths of the satellite with the following initial positions/velocities for one day function dx-at-sat ( tx) if r c" Ro, return ; ond% when colliding bnm6pos.n to solve a nonlinear d.e. on the orbit of a satellits global G Ne Re to 0; T 24*60-60: t T;N 2000 V20s (3071 3500 2000]: for iter t:length(v20s) x0 [x10 x20 v10 v201; tol. 1e-6; tR,xR) ode RK4tf, (t0 tri.xo,N: [t23s, x2351 ode23s (7777 (i) (x10?X1o) = (4.223 102.0)[m] and (x30. x40) s(h-no) = (0. 307 1)[m/s) (0. 3500)Im/s]. (0. 2000)[m/s]. (ii) (x10,X20) = (4.223 10. 0)[m] and (x30. xao) = (ulo-vo) = Run the program and check if the plotting results are as depicted in Fig. P6.5 (b) In Fig. P6.5, we see that the "ode23s()" solution path differs from the others for case (a) and the "ode45)" and "ode23s()" paths differ from the "ode_RK4()" path for case (ii). But, we do not know which one is more accurate. In order to find which one is the closest to the true solution, apply the two routines "ode45()" and "ode23s(" with smaller relative error tolerance of tol 10-6 to find the paths for the three cases. Which one do you think is the closest to the true solution among the paths obtained in (a)? but to warn the users of the danger of abusing them. With smaller number of steps (N) (ie.. larger step size), the routine "ode RK4 (0" will also deviate much from the true sulution. The MATLAB buili-in ruutines have too many good features to be mentioned here. Note that setting the parameters such as (cf) The purpose of this problem is not to compure the several MATLAB routines