Find an open interval I and a non-zero function g(x) on I such that (fg)' = f 'g' on I, or prove that they do not exist.
Solution
Answer: √(2x - 1)ex on (1, 2) (or many other intervals).
Trivial.
Put h(x) = ex2g(x). We require g'(x)/g(x) = 2x/(2x-1). Integrating: ln g(x) = x + 1/2 ln(2x - 1) + const, so g(x) = A √(2x-1) ex.
© John Scholes
jscholes@kalva.demon.co.uk
12 Dec 1998