Let's say I have an interval consisting of 10 elements and I calculate the Wronskian of the given functions. Out of the interval of 10 elements, substitution of 9 of the elements in the equation obtained after applying determinant(Wronskian) results in 0. But there is one number in the interval that doesn't lead the determinant value to zero. Would I still consider those functions as linear independent and also consider them to form the general solution for the ODE?

  • $\begingroup$ I don't understand the phrase "an interval consisting of 10 elements." Isn't an interval a thing like $a<x<b$, and would have infinitely many elements? $\endgroup$ – B. Goddard Mar 19 '19 at 18:37
  • $\begingroup$ Perhaps you mean: "at ten points belonging to the interval"? Then no, the functions cannot be solutions of any linear homogeneous ODE with the leading coefficient $1$ and the remaining coefficients continuous on the interval. But the functions are linearly independent: their linear dependence would result in their Wrońskian constantly equal to zero. $\endgroup$ – user539887 Mar 19 '19 at 18:56

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