Sturm-Liouville Separation theorem

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SUMMARY

The discussion centers on the necessity of the Wronskian being constant in the context of the Sturm-Liouville separation theorem. According to Abel's theorem, the relationship p(x)W[u1(x),u2(x)]=constant holds true, indicating that the Wronskian W[u1(x),u2(x)] must remain constant for solutions u1(x) and u2(x) of the Sturm-Liouville problem. The inquiry raises a potential exception where W[u1(x),u2(x)]=c/p(x), questioning whether this scenario can be disregarded. However, the consensus is that this special case does not invalidate the requirement for the Wronskian to be constant.

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kidsasd987
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Hi, I wonder why wronskian must be constant.

I know that p(x)W[u1(x),u2(x)]=constant, according to the Abel's theorem, but
wouldnt there a special case that W[u1(x),u2(x)]=c/p(x).

Then for this special case, W[u1(x),u2(x)]=/=c and satisfies Abel's theorem.
Is it ok to ignore this special case?
 

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this question is incomprehensible to me in connection with the accompanying text.
 

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