Few questions about solving DE

[STEMucator]

Homework Statement

See the image here : http://gyazo.com/de17b74fd351f26e2e361ae5d975b390

Homework Equations

The Attempt at a Solution

I'm a bit rusty at solving DEs since I was quite far ahead of the rest of the class and haven't practiced much. I'm just wondering if my thoughts about these are correct.

For the first one, I should solve the homogeneous system for a solution y_c and then use the method of undetermined coefficients to solve the non-homogeneous one. I believe my particular solution guess should be Y = Ax + b.

Same process for the second one, except my guess should be Y = Asin(2t) + Bcos(2t) + Ce^-4t.

For the third one, I'm pretty sure I solve the homogeneous system and then use the method of variation of parameters... or I could use something I figured out on my own time I'm pretty sure. Couldn't I just figure out a particular solution here by using :

$Y(t) = \sum_{m=1}^{n} y_m(t) \int \frac{g(t) W_m(t)}{W(t)}dt$ where y_m(t) is one of our known solutions to the homogeneous system, g(t) is our equation on the right hand side of the DE, W_m(t) is the determinant of W obtained by replacing the m^th column by (0, 0, 0, ..., 1) ( Cramer's Rule ), and W(t) is the wronskian of my known solutions to the homogeneous equation.

The last one looks ugly. Even though I can solve it using undetermined coefficients, I can tell it's going to get quite messy because if I do, my guess will be :

Y = (At⁵ + Bt⁴ + Ct³ + Dt² + Et + F)e^-t

Wouldn't abusing my formula from above again be more desirable here as it would probably reduce the algebra involved?

Thanks for the help in advance.