MATH 321: Final - Take Home Friday, December 22, 5 pm Show all appropriate work. Variables may represent any real number. 1. Update your Runge-Kutta function from the last exam so that it can solve a system of n coupled first order order differential equations (not necessarily linear). (a) Consider the IVP y (4) 8x2 y 000 y 00 + (y 0 )2 xyy 0 = cos x, y(0) = 1, y 0 (0) = 1, y 00 (0) = 0, y 000 (0) = 2. Use the function you created to solve the IVP using a step size of h = 0.01. Plot the solution on the interval [0, 0.75] and give a numeric value for the solution at x = 0.75. 2. I suggest you use MATLAB for this problem. If you do, hand in a printout of your command window. (a) Find the general solution to the differential equation 4y (5) 32y (4) + 77y 000 45y 00 + 27y 0 135y = 0. Hint: Look at the MATLAB help file for the roots function. (b) Solve the IVP 4y (4) 20y 000 + 17y 00 + 6y 0 + 45y = 0, y(0) = 0, y 0 (0) = 0, y 00 (0) = 0, y 000 (0) = 0