Did I do this Laplace transform correctly?

Question

1) $w'' + w = t^2 + 2$; $w(0) = 1$, $w'(0) = -1$

2) $s^2W(s) - sw(0) - w'(0) = \frac{2 + 2s^2}{s^3}$

3) $s^5W(s) - s^4w(0) - s^3w'(0) = 2 + 2s^2$

4) $ W(s) = \frac{2 + 2s^2 - s^3 + s^4}{s^5}$

5) $W(s) = 2\left(\frac{1}{s^5}\right) + 2\left(\frac{1}{s^3}\right) - \frac{1}{s^2} + \frac{1}{s}$

6) $w(t) = \frac{2}{4!}t^4 + t^2 - t + 1$

Anyone?

Answer 1

It seems that there is a defect in your solution. In fact, we have: $$s^2W(s)+1-s+W(s) = 2/s^3+2/s$$ instead which leads us to have: $$W(s)=\frac{2+2s^2-s^3+s^4}{s^3(s^2+1)}$$