I need to find a real general solution to y'' + 9y = 18x + 36 sin3x
Where are you stuck?
Can you solve y'' + 9y = 0?
Can you find a particular solution to y'' + 9y = 18x ?
To get a particular solution to y'' + 9y = 36 sin(3x) you will need to construct a particular solution of the form y = Ax [ sin(3x) + cos(3x) ]. Substitute and determine the value of A (tedious to be sure but not difficult).