R is the real line. Find all possible functions f: R → R with continuous derivative such that f(α)^{2} = 1990 + ∫_{0}^{α} ( f(x)^{2} + f '(x)^{2}) dx for all α.

**Solution**

Putting y = f(x) and differentiating the relationship gives 2y y' = y^{2} + y'^{2} or (y - y')^{2} = 0. So y = y' or y = -y'. Integrating, y = Ae^{x} or y = - Ae^{x}. But y(0) = ±√1990, so f(x) = ±√1990 e^{x}. The function is continuous, so we cannot "mix" the two solutions: either f(x) = √1990 e^{x} for all x, or f(x) = - √1990 e^{x} for all x.

© John Scholes

jscholes@kalva.demon.co.uk

1 Jan 2001