Hi the solution is y = Ae^(-t) - 1/2 sin(t) -1/2 cos(t)
You have the auxiliary equation k^2(k+1) = 0 where y = Ae^(kt) and k cannot be zero. So k= -1 and we have y=Ae^(-t)
Now for the complementary function ,try CF = a sin(t) + b cos(t). To find a and b
substitute in original equation to get
y= Ae^(-t) + asin(t) + b cos(t) differentiate to get
y' = -Ae^(-t) +a cos(t) - b sin(t) differentiate again to get
y'' = Ae^(-t) -a sin(t)+b cos(t)
We want y'' + y' = sin(t) so we can equate co-efficients in sin and cos to get a= -1/2 and b = -1/2
