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.
My suggestion was to differentiate, not integrate. That is, you have
Here's a link to Leibniz's Integral Rule. You'll need to differentiate again.
A quick check-in, then I'll get out of the way of the main discussion. The Leibniz rule. That is essentially the Fundamental Theorem mixed in with the chain rule, right? (I know that's only a brief description, I can fill in the details on my own if that is where it comes from.)
Thanks!
-Dan