y = f(x) is a solution of y'' + exy = 0. Prove that f(x) is bounded.
Solution
We have 2 e-x y' y'' + 2 y y' = 0. Integrating from 0 to k gives y(k)2 = y(0)2 - 2 ∫0k e-x y' y'' dx. Integrating by parts gives 2 ∫0k e-x y' y'' dx = e-x (y')2|0k + ∫0k (y')2 e-x dx = e-ky'(k)2 - y'(0)2 + ∫0k (y')2 e-x dx = A - y'(0)2, where A > 0.
Hence y(k)2 = y(0)2 + y'(0)2 - A < y(0)2 + y'(0)2, which establishes that y is bounded.
© John Scholes
jscholes@kalva.demon.co.uk
25 Jan 2002