the wronskian of two independent solutions of second order linear homogeneous Differential equation is never zero but can we say that wronskian of n independent solutions of n-th order linear homogenous Differential equation is never zero.?
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$\begingroup$ Wronskian is to check linear independence of the given solutions. …see en.m.wikipedia.org/wiki/Wronskian. It works for higher degree linear ode too. But essentially we need wronskian for non-homogenous linear ode $\endgroup$– Li Chun MinCommented Apr 9, 2017 at 17:26
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$\begingroup$ exponential function essentially does all the job for homogeneous ones $\endgroup$– Li Chun MinCommented Apr 9, 2017 at 17:27
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The result is also true for the $n$ linearly independent solution of the $ n^{th}$ order homogeneous linear differential equation. i.e., Wronskian of $n$ linearly independent solutions of $n^{th}$ order linear homogenous Differential equation is never zero.
For the proof of this general form theorem, we can see the following link:
https://people.clas.ufl.edu/kees/files/LinearIndependenceWronskian.pdf (Page 2, "3. Proof")
Also you can see:
"Ordinary differential equations" by Edward L. Ince (Page 116, Dover Publication Ins. (1978))
In this connection, I would like to mention you that you can found an new idea about the relation between the Wronskian with the linearly dependent and independent solutions in the footnote of the above mentioned book [Page 116].
It was Giuseppe Peano [G. Peano, Sur le déterminant Wronskien, Mathesis 9 (1889) 75-76] who first introduce the fact by giving an example showing that linearly independent functions may have a zero Wronskian i.e., The identical vanishing of the Wronskian is not a sufficient condition for the linear dependence of the n functions. Since then many research has been done on this matter and are still going on. You can find all these work in the Google. I am providing a link where you find some of those references
https://www.maa.org/press/periodicals/convergence/peano-on-wronskians-a-translation-references#BD-B2