Here is number one, Ill try and do some more...
I used a program so I could do this faster for you, basically for number 1, what you need to do is this:
To see if what they gave you is a solution: Find its derivatives (y , y1, y2 on my picture) and plug those in for y, y', and y''. Then simplify the expression. Upon doing so it should equal zero, which it does for part c.
EDIT: I should explain my work i think...as you can see I put the value in for y that they gave us...output 99
Next line, I set y1 equal to the derivative of y (so it is like y')....output 100
3rd line, I set y2 equal to the derivative of y1 (so its like y'', its just called something different because im using a program)...output 101
4th, I subbed in all of the values that we just found into the origional differential equation. I had the program simpify the answer.....output 102
Hopefully this makes sense?