- #1

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Why do we assume that the first term
cannot equal 0?

Thanks!

**c**in a frobenius series_{0}Thanks!

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In summary, the first term c0 in a Frobenius series cannot equal 0 because it is necessary for a0 to not vanish in order for the formal sum to converge and solve the differential equation. If a0 were to vanish, the result would not be interesting as the coefficient A could be any number.

- #1

- 190

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Why do we assume that the first term **c**_{0} in a frobenius series
cannot equal 0?

Thanks!

Thanks!

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- #2

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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.

- #3

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Thanks for the response. Why is it necessary though that a_{0} not vanish?

- #4

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IniquiTrance said:Thanks for the response. Why is it necessary though that a_{0}not vanish?

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

Aa

This implies Aa

We may assume a

Last edited:

In the Frobenius series method, the constant term c0 in the power series solution cannot be equal to 0. This is because if c0 is 0, then the resulting series becomes an infinite polynomial, which is not a valid solution for the differential equation.

No, c0 cannot be equal to 0 in any case for the Frobenius series method. This condition is necessary for the convergence of the series and for obtaining a valid solution for the differential equation.

The constant term c0 in the Frobenius series represents the initial condition of the differential equation. It is the value of the dependent variable at the initial point, and is used to determine the coefficients of the power series solution.

The value of c0 is determined by substituting the initial conditions of the differential equation into the power series solution. This allows for the coefficients of the series to be calculated and for the solution to be obtained.

Yes, c0 can be a complex number in Frobenius series. This is because the power series solution for a complex differential equation can also have complex coefficients, and c0 is just one of these coefficients.

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