Solved: Use the RungeKutta method for systems to approximate the solutions of the

Chapter 5, Problem 5.9.2

(choose chapter or problem)

Use the RungeKutta method for systems to approximate the solutions of the following systems of first-order differential equations, and compare the results to the actual solutions. a. u 1 = u1 u2 + 2, u1(0) = 1; u 2 = u1 + u2 + 4t, u2(0) = 0; 0 t 1; h = 0.1; actual solutions u1(t) = 1 2 e2t + t 2 + 2t 1 2 and u2(t) = 1 2 e2t + t 2 1 2 . b. u 1 = 1 9 u1 2 3 u2 1 9 t 2 + 2 3 , u1(0) = 3; u 2 = u2 + 3t 4, u2(0) = 5; 0 t 2; h = 0.2; actual solutions u1(t) = 3et + t 2 and u2(t) = 4et 3t + 1. c. u 1 = u1 + 2u2 2u3 + et , u1(0) = 3; u 2 = u2 + u3 2et , u2(0) = 1; u 3 = u1 + 2u2 + et , u3(0) = 1; 0 t 1; h = 0.1; actual solutions u1(t) = 3et 3 sin t + 6 cost, u2(t) = 3 2 et + 3 10 sin t 21 10 cost 2 5 e2t , and u3(t) = et + 12 5 cost + 9 5 sin t 2 5 e2t . d. u 1 = 3u1 + 2u2 u3 1 3t 2 sin t, u1(0) = 5; u 2 = u1 2u2 + 3u3 + 6 t + 2 sin t + cost, u2(0) = 9; u 3 = 2u1 + 4u3 + 8 2t, u3(0) = 5; 0 t 2; h = 0.2; actual solutions u1(t) = 2e3t + 3e2t + 1, u2(t) = 8e2t + e4t 2e3t + sin t, and u3(t) = 2e4t 4e3t e2t 2.