I've never heard this called "the polynomial method". And if your ODE was only slightly more complicated, then the result would not be a polynomial at all. But call it what you wish.
I'm not sure what is the problem, but it seems like you have given up without even starting. And since this is surely homework, I won't do your homework. That effort needs to come from you.
At this point, it looks like you need to figure out how to create a matrix. A loop would be sufficient. Or you could learn to use tools like diag, but a loop is just as esy. Or, for a 5x5 matrix, just write it out yourself. Many ways exist to build that matrix, and all are equally acceptable here.
Then you want to compute a determinant. Det will do.
Then you need to compute the roots of the result, which on this problem is indeed a polynomial. Solve will do.
You have it all written in front of you. Now you need to learn to use the tools in MATLAB to do that. And you even know what tools you need. So read the help for those tools if you need to, but then start by writing some code.
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