4.4 Undetermined Coefficients 
aka "informed guessing"

A) For an nth order CC ODE we know how to compute (Subscript[y, 1]),(Subscript[…y, n]) and find the complementary solutionb (Subscript[y, c])=(Subscript[c, 1])(Subscript[y, 1])+(Subscript[…c, n])(Subscript[y, n])
B) To complete the plan in 4.1 all we need for the general solution is any particular solution (Subscript[y, p]).
C) We can guess the form of (Subscript[y, p])from the RHS. 
D) All we have to do is work out the coefficients by pluggin into the equation.

({{RHS, (Subscript[Y, p])  Form}, {Sin[2x], A Sin[2x]+BCos[2x]}, {(x^2), A (x^2)+ B x + C}, {(E^(2x))Sin[3x], (AE^(2x))Sin[3x]+B (E^(2x))Cos[3x]}, {\[Placeholder], \[Placeholder]}, {\[Placeholder], \[Placeholder]}, {\[Placeholder], \[Placeholder]}, {\[Placeholder], \[Placeholder]}, {\[Placeholder], \[Placeholder]}, {\[Placeholder], \[Placeholder]}})

What could go wrong

The only thing that can go wrong is that one of our guessed terms could be in (Subscript[y, c]).  If this is the case it will vanish when we plug in.

Soln: Multiply by x till all the terms in your guess are out of the complementary solution.

What could get complicated

You could have lots of bits on the RHS.

Soln: Deal with the bits separetely and then add them up.