R is the real line. Find all possible functions f: R → R with continuous derivative such that f(α)2 = 1990 + ∫0α ( f(x)2 + f '(x)2) dx for all α.
Putting y = f(x) and differentiating the relationship gives 2y y' = y2 + y'2 or (y - y')2 = 0. So y = y' or y = -y'. Integrating, y = Aex or y = - Aex. But y(0) = ±√1990, so f(x) = ±√1990 ex. The function is continuous, so we cannot "mix" the two solutions: either f(x) = √1990 ex for all x, or f(x) = - √1990 ex for all x.
51st Putnam 1990
© John Scholes
1 Jan 2001