Please see the attached file for the complete solution.

find Y(t) for all t
Y''+2Y'+2Y = h(t)

Y(0) = 0

Y'(0) = 1

= {0 (less than or equal to) t ( less than)Pi and t ( greater than or equal to) 2 Pi
h(t)
= { 1 Pi (less than or equal to) t (less than) 2Pi

h(t) can be expressed as an unit step function

Then the Laplace transform of h(t) is

Assume the Laplace transform of y(t) is Y(s)

Then

So take the Laplace transform on both sides of the equation

(1)

First we ignore and , just concentrate on .
Now for

That is, the above equation has 2 complex roots. So

When B and B* are complex conjugate.

So

Then the inverse transform of the above function is
(2)

Now remember in equation (1), we have
Then the inverse Laplace transform of is

Then the inverse Laplace transform of is

So the inverse Laplace transform of Y(s)

So the solution is