Sorry,
@Mark44 , I meant T by t - 1, not r - 1.
Here are the steps I did:
y'' + 4y = 8t^2 * [1 - u(t-5)] = 8t^2 - 8t^2 * u(t-5)
ỹ'' + 4ỹ = 8(T + 1)^2 - 8(T + 1)^2 * u(T - 4)
ỹ'' + 4ỹ = 8(T^2 + 2T + 1) - 8(T^2 + 2T + 1) * u(t - 4)
I use Laplace on both sides and get:
s(s + 4)Ỹ - s - scos(2) + 2sin(2) - 8 - 4cos(2) = 16/s^3 + 16/s^2 + 8/s - 8ℒ(T^2 * u(t - 4)) - 16ℒ(T * u(t - 4)) - 8ℒ(u(t - 4))
s(s + 4)Ỹ - s - scos(2) + 2sin(2) - 8 - 4cos(2) = 16/s^3 + 16/s^2 + 8/s - 8e^(-4s) * (2/s^3 + 8/s^2 + 16/s) - 16e^(-4s) * (1/s^2 + 4/s) - 8e^(-4s) * 1/s
s(s + 4)Ỹ - s - scos(2) + 2sin(2) - 8 - 4cos(2) = (16 + 16s + 8s^2) / (s^3) - e^(-4s) * (16 - 80s - 200s^2) / (s^3)
Then I solve for Ỹ on the left, and get:
Ỹ = (16 + 16s + 8s^2) / [(s^4)(s+4)] - e^(-4s) * (16 - 80s - 200s^2) / [(s^4)(s+4)] + (s + scos(2) - 2sin(2) + 8 + 4cos(2)) / [s(s+4)]
Now I use partial fraction decomposition on all 3 terms:
first term) (16 + 16s + 8s^2) / [(s^4)(s+4)] = (As^3 + Bs^2 + Cs + D) / (s^4) + E / (s + 4)
I get A = -5 / 16, B = 5 / 4, C = 3, D = 4, E = 5 / 16second term) (16 - 80s - 200s^2) / [(s^4)(s+4)] = (As^3 + Bs^2 + Cs + D) / (s^4) + E / (s + 4)
I get A = -179 / 16, B = 179 / 4, C = 21, D = -4, E = 179 / 16third term) (s + scos(2) - 2sin(2) + 8 + 4cos(2)) / [s(s+4)] = A / s + B / (s + 4)
I get A = 2 - (1/2)sin(2) + cos(2), B = (1/2)sin(2) - 1Now I put it all together:
Ỹ = (-5 / 16s) + (5 / 4s^2) + (3 / s^3) + (4 / s^4) + (5 / 16(s+4)) + e^(-4s) * [(-179 / 16s) + (179 / 4s^2) + (21 / s^3) + (-4 / s^4) + (179 / 16(s+4))]
+ [2 - (1/2)sin(2) + cos(2)] / s + [(1/2)sin(2) - 1] / (s + 4)
(At this point I think I have already made a mistake somewhere, because it looks too messy to be correct)
Now I use inverse Laplace on both sides and get:
ỹ = (- 5 / 16) + (5/4)T + (3/2)T^2 + (2/3)T^3 + (5/16)e^(-4T) - (179/16)u(T - 4) + (179/4)(T-4)u(T - 4) + (21/2)(T-4)^2 * u(T - 4) - (2/3)(T-4)^3 * u(T - 4)
+ (179/16)e^(-4T + 16) * u(T-4) + [2 - (1/2)sin(2) + cos(2)] + [(1/2)sin(2) - 1]e^(-4T)
Now I simplify, and substitute T with t - 1, and get:
y = -5/16 + (5/4)(t-1) + (3/2)(t^2 -2t + 1) + (2/3)(t^3 -3t^2 +3t -1) + (5/16)e^(-4t+4) + 2 - (1/2)sin(2) + cos(2) + (1/2)sin(2)e^(-4t+4) - e(-4t+4) + (179e^20 / 16)e^(-4t)
+ [5323/48 - (441/4)t \ (41/2)t^2 - (2/3)t^3]u(t-5)
Thanks a lot for the feedback.
P.S.
The final expression, according to Kreyszig's "Advanced Engineering Mathematics, 9th edition", should be:
cos(2t) + 2t^2 - 1, if 0 < t < 5
cos(2t) + 49cos(2t - 10 + 10sin(2t - 10), if t > 5