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 − (2t2 + 1)e2t , u1(0) = 1;
u'2 = 4u1 + u2 + (t2 + 2t − 4)e2t , u2(0) = 1; 0 ≤ t ≤ 1; h = 0.2;
actual solutions u1(t) = 1/3 e5t - 1/3 e−t + e2t and u2(t) = 1/3 e5t + 2/3 e−t + t2e2t .
b. u'1 = −4u1 − 2u2 + cos t + 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) = 2e−t − 2e−2t + sin t and u2(t) = −3e−t + 2e−2t .
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) = cos t + sin t − et + 1, u2(t) = −sin t + cos t − et , and u3(t) =
−sin t + cos t.
d. u'1 = u2 − u3 + t, u1(0) = 1;
u'2 = 3t2, u2(0) = 1;
u'3 = u2 + e−t , u3(0) = −1; 0 ≤ t ≤ 1; h = 0.1;
actual solutions u1(t) = −0.05t5 + 0.25t4 + t + 2 − e−t , u2(t) = t3 + 1, and u3(t) = 0.25t4 + t − e−t .