I was told to apply the method of Laplace transforms to solve for x.
the displacement $x:=x(t)$ of a spring-mass satisfies the IVP: $$\ddot{x}+4\pi^2x=\sum_{k=1}^3 2\piδ(t-k)$$ for all $t>0$, and $$x(0)=7$$ $$\dot x(0)=0$$ The summation on the right represents 3 impulsive forces applied to the spring-mass system externally at time $t=1,2,3$.
I tried expanding the summation on the RHS, giving $$\ddot{x}+4\pi^2x=6\piδ(t-2)$$ (which was differnet from Cesareo suggested). Then, applying Laplace transformation in both sides resulting in $$L(x)=\frac{6\pi e^{-2s}+7s}{s^2+4\pi^2}$$ I am not sure if i get the Laplace transformation right. May I know how can I proceed to solve for $x$? Shall I just go for the inverse Laplace transforms? Thank you so much.