I am not sure what you are doing.

There is a Generalized Variation of Parameters.

But again it involves finding all 3 independent solution to,

y'''+y'=0

Which is,

z''+z=0 (after reduction of order).

Hence,

z=y'=C_1sin(t)+C_2cos(t)

But then,

y=C_1sin(t)+C_2cos(t)+C_3

Thus,

y_1=sin(t)

y_2=cos(t)

y_3=1

Are three linearly independent solutions.

Now! You can use the generalized variation of parameters techinique for linear differencial equations of order 3.