How was this equation able to be manipulated in order to solve the Laplace transform?

Question

I have this equation that I'm supposed to take the Laplace transform of, but I'm not sure how and I also don't understand the two solutions I found online for it: $$y''+y=t-(t-4)u(t-2)$$ I know how to solve the rest of it, but the $(t-4)u(t-2)$ portion I'm not sure about. A friend told me to put the equation into this form: $${((t-2)-2)u(t-2)}$$ this left me with $$e^{-2s}L[(t-2)](s)$$ I understand this works, but then I'm not sure how to take the Laplace of $(t-2)$. Another source told me to change the equation to be in this form: $$y''+y=t-(t-2)u(t-2)+2u(t-2)$$ I know how to solve this, but I don't understand how the person got this solution. Why is this allowed?

Answer 1

All that was done was some algebraic manipulation; surely we agree that

$$(t - 4) = (t - 2 - 2)$$

so we have

$$\begin{align}t−(t−4)u(t−2) &= t - ((t-2)-2)u(t-2) \\ &= t - (t-2)u(t-2) - (-2\cdot u(t-2))\\ &= t - (t-2)u(t-2) + 2u(t-2)\end{align}$$