Frobenius Series: The Significance of Non-Zero First Terms

[IniquiTrance]

Why do we assume that the first term c₀ in a frobenius series

cannot equal 0?

Thanks!

 

[Ben Niehoff]

Latex seems to be misbehaving, so I'll write in plain text:

In the Frobenius substitution, the x dependence of the first term is already factored out:

y(x) = x^r Sum (a_k x^k)

So, the first term in the series is actually

a_0 x^r

and when we plug the series into the differential equation, the question we are asking is "What is the smallest r for which a_0 does not vanish?" The answer is given by the indicial equation.

After solving the indicial equation for r, we are then equipped to ask the next question: "Given that a_0 does not vanish, can I find some sequence a_k such that my formal sum converges and solves the differential equation?"

 

[IniquiTrance]

Thanks for the response. Why is it necessary though that a₀ not vanish?

 

[matematikawan]

Thanks for the response. Why is it necessary though that a₀ not vanish?

Let say the first term that we obtained on substituting the Frobenius series into the DE as

Aa₀x^r + ... is identically zero.

This implies Aa₀=0.
We may assume a₀ to be zero or nonzero. But if it is zero then A can be any number. Not an interesting result.