Answer: Use the Runge-Kutta method for systems to approximate the solutions of the

Chapter 5, Problem 1

(choose chapter or problem)

Use the Runge-Kutta 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 = 3u1 + 2u2 (2t 2 + 1)e2t , u1(0) = 1; u 2 = 4u1 + u2 + (t 2 + 2t 4)e2t , u2(0) = 1; 0 t 1; h = 0.2; actual solutions u1(t) = 1 3 e5t 1 3 et + e2t and u2(t) = 1 3 e5t + 2 3 et + t 2e2t . b. u 1 = 4u1 2u2 + cost + 4 sin t, u1(0) = 0; u 2 = 3u1 + u2 3 sin t, u2(0) = 1; 0 t 2; h = 0.1; actual solutions u1(t) = 2et 2e2t + sin t and u2(t) = 3et + 2e2t . c. u 1 = u2, u1(0) = 1; u 2 = u1 2et + 1, u2(0) = 0; u 3 = u1 et + 1, u3(0) = 1; 0 t 2; h = 0.5; actual solutions u1(t) = cost + sin t et + 1, u2(t) = sin t + cost et , and u3(t) = sin t + cost. d. u 1 = u2 u3 + t, u1(0) = 1; u 2 = 3t 2, u2(0) = 1; u 3 = u2 + et , u3(0) = 1; 0 t 1; h = 0.1; actual solutions u1(t) = 0.05t 5 + 0.25t 4 + t + 2 et , u2(t) = t 3 + 1, and u3(t) = 0.25t 4 + t et . 2. Use t