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There are so many methods for checking linear independency of functions. But why do we require this? I mean what is the use to find out whether functions or solutions are linear independent or not.

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Linear independence of solutions is useful for describing the entire solution space. Typically, with an ODE you have a finite-dimensional space of solutions, and to describe this solution space you want to supply a list of solutions $\{f_1,\ldots,f_n\}$ that will be sufficient to describe any solution to the equation via a linear combination $\alpha_1f_1 + \cdots \alpha_nf_n$. It's helpful in this process to ensure that the functions $f_1,\ldots,f_n$ are linearly independent, since otherwise we would effectively be throwing in redundant solutions that do us no good in this regard. Thus, ensuring that the set is linearly independent guarantees that we have exactly as many functions as we need and no more.

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    $\begingroup$ Moreover throwing in "redundant" solutions probably means we do not have the full general solution, and cannot satisfy all initial conditions. $\endgroup$ – JP McCarthy Feb 26 at 7:50

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