Here, let's take a look at sums of the fundamental components and/or products of the fundamental components. To do this we'll require the following fact.

Fact- Undetermined Coefficients

If Y_P1(t) is a particular solution for

 y′′ + p (t) y′ + q (t ) y = g1 (t)

 And if Y_P2(t) is an exact solution for

y′′ + p (t ) y′ + q (t) y = g2 (t)

So Y_P1(t)+ Y_P2(t) is an exact solution for

y′′ + p (t ) y′ + q (t ) y = g1 (t )+ g2(t)

This fact can be utilized to both find exact solutions to differential equations which have sums in then and to write down guess for functions which have sums in them.