y=Acos2x + Bsin4x

Question

How to eliminate the arbitrary constant a and b

Answer 1

Make some constants , a, b and c, such that ay''+by'+cy=0 to eliminate A and B from your expression. Plugging in y'', y' and y and collecting like terms, you can set up a system of equations that will accomplish this (one of a, b, c will obviously be a free variable that you can just pick).

he actual reason for it, however, is that a, b and c are not unique: You can multiply ay''+by'+cy=0 by any constant and the expression will still be true, meaning that if (a,b,c) does the job so does (ka,kb,kc). By picking one of the constants, you force the other two to take a definite value.