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I am a bit stuck in this question that I found in my textbook - "Show that the Wronskian of the functions $x $, $x^2 $, and $x^2\log x$ is non zero. Can these be independent solutions of an ordinary differential equation? If so, determine the differential equation."

My question is that if the Wronskian turns out to be non zero won't the functions automatically be independent solutions of an ODE?

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  • $\begingroup$ And how do you determine such an ODE? $\endgroup$
    – John B
    Apr 29 '16 at 10:52
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Well, if the Wronskian is nonzero (except at isolated points), then the functions are linearly independent (see Wikipedia). However, that does not automatically mean that they are the solutions of the same ODE.

You could try to construct an ODE of which these functions are solutions, though. To do that, you must first ask yourself what the order of that ODE should be. Then, I would suggest you try to find a linear ODE of that order which has the functions $x$, $x^2$ and $x^2 \log x$ as solutions.

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