I have just been solving some systems of ODEs to find the nomal mode solutions. Something about it has been bugging me though. In a simple case where we have a system of two linear ODEs representing a two-mass spring system, we assume that the solution is a normal mode and so find a general solution using eigenvalues and eigenvectors and get a vector solution involving a linear combination of four terms each multiplied by an arbitary constant. No problem there. The part that is worrying me though is this: We assumed that the solution was a normal mode; so what if it wasn't? I.e. do non-normal-mode solutions exist? If they do then why have we got four arbitary constants in our normal-mode solution - surely if a soulution of this system involves four constants then it describes the most general solution. Also, is it technically still called a normal mode if we combine two normal modes to get another solution? Any guidance on this matter would be great thanks.