Solving String Vibration Using Integral Transform

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$$U_{tt} - c^2 U_{xx}= -g$$ where

BC: $U_{x}(0,t)=a\sin(ωt)$

IC: $U(x,0)=0$, $U_{t}(x,0)=0$

where $c, g, A$ and $ω$ are positive constants

Normally I wouldn't post for help here but I am absolutely desperate. I know how to solve this using the eigenvalue expansion method but I am absolutely terrible at Laplace and Fourier Transforms (Which i think i need to use one of the two).