I understand that if $y_1$ and $y_2$ are solutions, then $Ay_1+By_2$ is also a solution, and I understand by $A$ and $B$ can take any number. But why are there only 2 things (here $y_1$ and $y_2$, I encountered the word 'degrees of freedom' alluding to the same thing in Feynman's lectures), and we can be sure that there is not a $y_3$, that is not a multiple of $y_2$ of $y_2$, hiding away?
I assume that this problem generalises to having n independent solutions for an nth order linear homogeneous equation.
If it helps in the writing of your answers, am very much a beginner in differential equations.