y = f(x) is a real-valued solution (for all real x) of the differential equation y'' - 2y' + y = 2ex which is positive for all x. Is f '(x) necessarily positive for all x? y = g(x) is another real valued solution, which satisfies g'(x) > 0 for all real x. Is g(x) necessarily positive for all x?
(a) no, (b) yes.
We know that the general solution of y'' - 2y' + y = 0 is A ex + B x ex. So the particular solution to y'' - 2y' + y = 2ex must be of the form C x2 ex, and, indeed, differentiating, we find C = 1. So the general solution is f(x) = x2 ex + A ex + B x ex. That is all pure book work.
It is now convenient to rewrite this in a slightly different form: f(x) = ( (x + a)2 + b ) ex, where a and b are still arbitary constants. Then f '(x) = ( (x + a + 1)2 + b - 1) ex. The answers are now obvious. f(x) is always positive iff b > 0, and f '(x) is always positive iff b > 1. So the answer to (a) is No, and the answer to (b) is Yes.
48th Putnam 1987
© John Scholes
7 Oct 1999