Use the Runge-Kutta for Systems Algorithm to approximate the solutions of the following higherorder differential equations, and compare the results to the actual solutions.
a. y'' − 3y' + 2y = 6e−t, 0≤ t ≤ 1, y(0) = y'(0) = 2, with h = 0.1;
actual solution y(t) = 2e2t − et + e−t .
b. t2y'' + ty' − 4y = −3t, 1≤ t ≤ 3, y(1) = 4, y' (1) = 3, with h = 0.2;
actual solution y(t) = 2t2 + t + t−2.
c. y''' + y'' − 4y' − 4y = 0, 0 ≤ t ≤ 2, y(0) = 3, y' (0) = −1, y'' (0) = 9, with h = 0.2;
actual solution y(t) = e−t + e2t + e−2t .
d. t3y''' + t2y'' − 2ty' + 2y = 8t3 − 2, 1 ≤ t ≤ 2, y(1) = 2, y' (1) = 8, y'' (1) = 6, with h = 0.1; actual solution y(t) = 2t − t−1 + t2 + t3 − 1.