@Neel What you have here is so far removed from what would be the right approach to the solution of the problem, that it leaves little to salvage. In my earlier message I provided a general outline for solving the forced vibration problem. What you have in your worksheet does not follow that outline at all. The entire worksheet needs to be discarded and started over again.
But you should not feel bad about that. My outline assumes some familiarity with orthogonal basis functions generated by solutions of boundary value problems, and how to expand an arbitrary function in such a basis. It appears that you have not seen that material, so I wouldn't blame you for that.
I can provide the somewhat long and messy solution to the problem which you are attempting to solve, but without the proper mathematical background it won't be understandable.
Why are you interested in this problem? If it is a student project, you should really begin with learning the basic mathematical ideas behind what is called eigenfunction expansions. Furthermore, the beam problem is one or two steps more advanced than a beginner problem in this subject, which would be:
First: The heat equation:
w_t= w_xx + f(x,t),
w(x,0) = g(x),
Second: The wave equation:
w_tt = w_xx + f(x,t),
w(x,0) = g(x), w_t(x,0) = h(x).
You need to be able to confortably handle these before attempting the beam equation. Do you have a textbook that shows how to solve these problems? Learning that will be the first in your progression. By the way, notice the initial conditions g(x), and h(x). You will need to specify initial conditions to solve your beam problem as well. I didn't see initial conditions in your worksheet.