So i have been trying to solve this Laplace transform for some time now. I have asked my assistant teacher and he also was not able to solve it, so i will try here. problem: $$ y'''- y = -ye^{2t}, \ \ \ \ \ y(0) = 0 , \ \ y'(0)=-1, \ \ y''(0)=-3. $$ Have tried some different approaches with getting the same $Y(s)$ value and also tried to Taylor expand $Y(s-2)$ and also tried it this way:
$$ \mathscr{L} \{ y''' \} = s^3Y(s)-s^2(0)-s(-1)+3 \\ \mathscr{L} \{ y\} = Y(s) \\ \mathscr{L} \{ ye^{2t}\} = Y(s-2) $$
$$ s^3Y(s)+s+3-Y(s)=Y(s-2) \Longrightarrow Y(s)(s^3-1)=Y(s-2)-s-3 $$
$$ Y(s) = \frac{Y(s-2)}{s^3-1}-\frac{s}{s^3-1}-\frac{3}{s^3-1} $$ And from here on i have also tried to get something that makes sense. Problem is found here on a exercise sheet under the ["Recommended Exercises"] https://wiki.math.ntnu.no/_media/tma4130/2021h/exercise_04.pdf. And since its under recommended exercises we dont have a solutions manual for it...