Question

If f and g are two functions for which f' = g and g' = f for all x, then prove that
f^2 - g^2 must be a constant.

Answer 1

ff' = fg
gg' = gf

ff' = gg'
2f f' - 2g g' = 0
f^2 - g^2 = c

Answer 2

I don't really get why f^2 - g^2 would be a constant.

Answer 3

take integrals across the equation

2ff' - 2gg' = 0
∫2f df - ∫2g dg = 0
(f^2 + c1) - (g^2 + c2) = 0
f^2 - g^2 = -(c1+c2) = c