]My question is how do you from a proper guess at the particuliar solution of a differential equation for example.

Say this is the right hand of the equation

I know it helps to break it up into seperate parts, but I am working with a book solution here and my guess are just off it seems from the books proper guess.

My guess for

Book's guess

and the for the term of 4 the book's guess was

How do you properly determine a guess like this?