For a certain DE, how can we be certain how many solutions there are? Also, how can we determine the general set? For example, for the linear, second order, homogeneous DE, you may have ay'' + by' + cy = 0  and then the characteristic equation of the form ar^2 + br + c can be found, and thus the corresponding r values to yield the general solution of the form: c1y1 + c2y2 where y1 = e^(r1t) and y2 = e^(r2t). For this, you find the Wronskian, and if non-zero, then you know these solutions are independent. That's great. And if anything up to here is incorrect, please correct me. But my question is: how are we CERTAIN c1y1 + c2y2 contains every possible solution to ? I understand this is a solution, but don't see why this encompasses every single solution. Likewise (I assume it is corollary), why does an nth order, linear, homogeneous DE have n terms in the general solution?