Let y = Acost + Bsint
y' = -Asint + Bcost
y" = -Acost - Bsint
Since y" + 8y' + 17 = cost, we have:
(-Acost - Bsint) + 8(-Asint + Bcost) +17(Acost + Bcost) = cost
(16A + 8B)cost + (-8A + 16B)sint = (1)cost + (0)sint
So, 16A + 8B = 1
-8A + 16B = 0
Multiply the second equation by 2:
16A + 8B = 1
-16A + 32B = 0 So, 40B = 1. Therefore, B = 1/40 and 16A + 1/5 = 1. So, A = 1/20.
Particular solution is y = (1/20)cost + (1/40)sint.