Isn't the characteristic equation or Then you'd be trying to find the fourth roots of .
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.