The question:
Write down the form of the particular solution you should try when solving the non-homogeneous differential equation
My answer:
Does this look correct? Thanks.
What you've written is the Characteristic Solution, not the Particular Solution.
Since your RHS of the DE is of the family , you would normally choose as a particular solution.
But since appears in your Characteristic Solution, you would then normally choose as a particular solution.
But since appears in your Characteristic Solution, you would then normally choose as a particular solution.
So I agree with the OP's choice of .