For a certain DE, how can we be certain how many solutions there are? Also, how can we determine the general set?(adsbygoogle = window.adsbygoogle || []).push({});

For example, for the linear, second order, homogeneous DE, you may have ay'' + by' + cy = 0 [1] 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 [1]? 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?

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# Determining # of Solutions for DEs

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