you are correct so far. but this problem requires stamina. keep expanding and you will need to use a similar identity again, but just use it. at the end, you will have a string of terms, each of which you can integrate easily

just find x''(t) and plug it into the equation given (plug in x(t) as well). if the equation makes sense (that is, both sides simplify to the same thing), it is a solution

Problem 2: Suppose that A and B are constants. Show that the function x(t)=Acoshkt+Bsinhkt is a solution of the differential equation

Thank you!