Isn't the characteristic equation or Then you'd be trying to find the fourth roots of .
I suspect you are misreading your text book. If i and -i are roots of the characteristic equation, then, yes, and are solutions and the general solution will be of the form (I am dropping the and since they are not relevant to this point).
Now, as you know, and (because cos(x) is an even function and sin(x) is an odd function). That gives and, defining C to be A+ B and D to be (A- B)i, this is Ccos(x)+D sin(x).
That is, it is the and together that give both trig functions. It is NOT the case that alone gives "cos(x)" and alone gives "sin(x)".
Hey- Thanks for your time. Unfortunately, I'm not misreading. And it's actually not a text book, it has the word "Dummies" in the title, and that might be part of the problem. I'm going to forge on and hope I can learn something. I'm going to see if I can arrive at the correct general solution to that DE.