Your first equation is applying a force at all points $0<x<L$ of the string rather than at one end. It therefore couples to all of modes with $n$ an odd number. You will get some motion in all of these modes from this equation.

If you want to mimic the experimental condition you will need a different equation--- one that will depend on exactly how you apply the external force. If you shake the end transvesely, you can keep $$ u_{tt}-c^2 u_{xx}=0 $$ but solve with boundary condition $$ u(x=0,t)= A \sin \omega t. $$

Your first equation is applying a force at all points $0<x<L$ of the string rather than at one end. It therefore couples to all of modes with $n$ an odd number. You will get some motion in all of these modes from this equation.

You force does not couple to the $n=2$ mode $$ u(x,t)= \sin(2\pi x/L) \sin(2\pi ct/L) $$ because $$ \int_0^L \sin (2\pi x/L) \sin(2\pi ct/L)dx=0. $$ If you have solved correctly you should get no motion from your equation in this case .

If you want to mimic the experimental condition you will need a different equation--- one that will depend on exactly how you apply the external force. If you shake the end transvesely, you can keep $$ u_{tt}-c^2 u_{xx}=0 $$ but solve with boundary condition $$ u(x=0,t)= A \sin \omega t. $$