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'' − 2y' + y = tet − t, 0≤ t ≤ 1, y(0) = y'(0) = 0, with h = 0.1;
actual solution y(t) = 1/6 t3et − tet + 2et − t − 2.
b. t2y'' − 2ty' + 2y = t3 ln t, 1≤ t ≤ 2, y(1) = 1, y' (1) = 0, with h = 0.1;
actual solution y(t) = 7/4 t + 1/2 t3 ln t - 3/4 t3.
c. y''' + 2y'' − y' − 2y = et, 0≤ t ≤ 3, y(0) = 1, y' (0) = 2, y'' (0) = 0, with h = 0.2;
actual solution y(t) = 43/36 et + 1/4 e−t - 4/9 e−2t + 1/6 tet .
d. t3y''' − t2y'' + 3ty' − 4y = 5t3 ln t + 9t3, 1≤ t ≤ 2, y(1) = 0, y' (1) = 1, y'' (1) = 3,
with h = 0.1; actual solution y(t) = −t2 + t cos(ln t) + t sin(ln t) + t3 ln t.