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Am I gauranteed that these two solutions will be linearly independent or do I need to verify with the Wronskian each time?

Also, can I think of linearly independent solutions as a "basis" like I do in linear algebra? It seems like it because its my understanding that any solution for the ODE is simply a linear combination of solutions in this "basis"

Also, can we tell the dimensions of this basis based on the order of the ODE? Like, a second order ODE has two solutions in its basis, a third order has 3, a first order has one, etc.

Thanks!