Suppose that f'(x) = -g(x) and g'(x) = f(x) for all real x?

Question

Suppose that f'(x) = -g(x) and g'(x) = f(x) for all real x
a. Prove that f^2(x) + g^2(x) = C for some constant C.
b. Suppose that f(0) = 0 and g(0) = 1. What is C?

Answer 1

a. d/dx(f^2(x) + g^2(x)) = 2f(x)f'(x) + 2g(x)g'(x) = 0

b. 0^2 + 1^2 = 1