linear independence of solutions
Two solutions are useful as building blocks only if they genuinely point in different directions — if one is not just a rescaled copy of the other. If your two 'basic' solutions were really the same shape, say y2 = 3 y1, then combining them could never give you anything new, and you would be stuck with a one-parameter family when you need two. Linear independence is the precise way of saying the two solutions are truly different.
Two functions y1 and y2 are linearly dependent on an interval if one is a constant multiple of the other — equivalently, if there are constants c1 and c2, not both zero, with c1 y1(x) + c2 y2(x) = 0 for all x. They are linearly independent if no such relation exists: the only way to make c1 y1 + c2 y2 vanish identically is c1 = c2 = 0. For two functions there is a quick mental test — independent means the ratio y2/y1 is not constant. For solutions of the same linear equation there is an even sharper tool: they are independent exactly when their Wronskian (the determinant y1 y2' - y2 y1') is nonzero at some point of the interval.
Independence is the dividing line between a useful pair and a redundant one. To get the general solution of a second-order equation you need two independent solutions; a dependent pair spans only a one-dimensional slice of the two-dimensional solution space and leaves half the solutions unreachable. So whenever you hand-find two solutions, the first thing to check is that they are not secretly proportional.
For y'' + y = 0, sin(x) and cos(x) are independent: their ratio tan(x) is not constant, and their Wronskian is sin(x)(-sin x) - cos(x)(cos x) = -1, never zero. By contrast sin(x) and 2 sin(x) are dependent — ratio 2, Wronskian zero — and span only one dimension.
A nonzero Wronskian certifies independence; a constant ratio (Wronskian zero) exposes a redundant pair.
For functions in general, a zero Wronskian does not prove dependence — there are independent functions with vanishing Wronskian. But for two solutions of the SAME linear ODE on an interval, the Wronskian test is exact, thanks to Abel's identity.