(20 pts) Solve the IVP: y'''' - y = u1 - u2, y(0) = 0, y'(0) = 0, y''(0) = 0, y'''(0) = 0.
For this problem we shall be using the Laplace transform. Because of the vanishing initial conditions, the equation is getting transformed into
Let's ignore the numerator for a moment and find the inverse Laplace transform of G(s) = 1/s(s4 - 1). The partial fractions is the way to go. Since
Reduction to the common denominator gives:
Let's compare the coefficients of various powers of s on both sides:
4: 0 = A + B + C +E.
3: 0 = A - B + D.
2: 0 = A + B - C.
1: 0 = A - B - D.
0: 1 = -E
The last equation gives E = -1. From the second and the fourth equations, A - B = 0. So that A = B. From the first and the third equations, A + B = -E/2 = 1/2. Conclude that
This is the Laplace transform of the function
The solution is the difference of two translations of g:
y(t) = u1(t)·g(t - 1) - u2(t)·g(t - 2).