Let's look at the first one:

(1)

The differential operator

annihilates

and

annihilates

hence the operator:

annihilates

.

Thus, applying

to both sides of (1) gives us:

(2)

The characteristic roots are then:

where

is of multiplicity 4 and

are of multiplicity 2, and so the general solution to (2) is given by:

(3)

Now, recall that a general solution to (1) is of the form

. Since every solution to (1) is also a solution to (2), then

must have the form displayed on the right-hand side of (3). But we recognize that:

and so there must exist a particular solution of the form:

Now it is a matter of using the method of undetermined coefficients to determine the particular solution that satisfies (1).

So, what you need to do is compute

and

, substitute these into (1), and solve the resulting linear system that arises when you equate coefficients. Can you proceed?