suppose that f(x) is a twice differentiable function which satisfies f"(x) = -f(x) for all x.?

Question

suppose that f(x) is a twice differentiable function which satisfies
f"(x) = -f(x) for all x. We show that f(x) = Acosx + Bsinx for certain constants A and B as follows:


a) show that f(x)cosx - f'(x) sinx is constant
b) show that f(x) coxx + f'(x) sinx is constant
c)use the first 2 parts to complete the solution.

Answer 1

Well, for a and b, how do you show something is a constant???

Differentiate it and check that the result is 0.

The product rule is your friend, and substitute -f for f" as needed.

c would then be simple algebra.

Only -- you made a serious typo, and I don't mean cox for cos. So correct that before you get to work. You'll find that when you cancel out the f' term, the f term will have a coefficient of (sin^2 + cos^2). And you know what that equals!