Solving 2nd Order ODE w/Laplace Transforms + Heaviside

Question

This is a similar problem to the one I posted earlier with some differences.

Attempt at solution:

1. Write $g(t)$ as a heaviside function.
2. Take Laplace transform of LHS and RHS.
3. Solve for Y.
4. Take inverse laplace of Y. This gives me a function $y(t) = something$.

How do I change my answer that I get as $y(t)$ into the form they want in the image?

How do I write $g(t)$ as a Heaviside function? I get $g(t) = t + (2\pi-t)u$, where $u$ is the Heaviside $u$ of $2\pi$.

Answer 1

$g(t) $ should have the form in terms of Heaviside function

$$g(t) = t u( 2\pi - t) + 2\pi u( t-2\pi).$$