Note that we do not need to use two constants of integration because a single constant is equivalent. Why is this so? I have highlighted this on the Wikipedia article that caused the confusion. http://awurl.com/WYS0RD58q
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The constants have not been assigned a value. So, since C1-C2 is just another constant without an assigned value, we can refer to it as C still.
Jhow is correct, but moreover--we know that C_1,C_2...C_(n-1),C_n are all constants, and thus when added, subtracted etc will still be SOME constant, and thus can simply be constrained to a single constant, since in the end--if we need to know it (and have an inital value) we will solve for the FINAL constant.
yup! I agree ... constants are used to represent the general equation. eg
f(x)+C represents a general equation while f(x)+5 represent a particular solution.
for example we have, f(x)+5 = g(y) + 10 as particular solution from your differential equaion, then f(x) = g(y)+5 is equivalent solution, further more constant are zero on differentiation. so, you can represent C1-C2 or whatever by a single constants.
for example integrate X^2 dx + X dx + 5dx ... here you should get 3 constants, but sum of three constant is a constant which is used to represent a general term. so one if fine.