Your DE is a second order linear homogeneous equation. One route you could take is the following:
1. Set up what's called the "characteristic equation." This is done by replacing your DE with the polynomial
(if you haven't seen this before and need more details let me know).
2. Solve the above for Note: The two solutions for will be a complex numbers.
3. The general solution to your DE is then
where is the real part of and is the imaginary part of . See Equation 11 of http://www.stewartcalculus.com/data/...arEqns_Stu.pdf
4. Now use the initial conditions to determine and .
5. To prove part c) I would suggest visiting http://www.mathcentre.ac.uk/resource...a-alphaetc.pdf. There is a nice outline showing how to express the sum of sine and cosine as one cosine function.
Does this help sort out the confusion?