Why don't you just differentiate the y there twice, plug into the DE, and see if you get equality?
Show that y'' − y = F(t) can be written as y= Ae^t+Be^-t + \int sinh(t-\tau )F\tau d\tau
Obviously, the first is the complementary solution which is easily obtained. The second part I have no idea about. Tried using a variation of parameters but how to get that integral I don't know.