Difference between power series method and Frobenius method

Question

There is the power series method for solving ordinary differential equations: one looks for solutions of the form $\sum c_n x^n$, and derives algebraic relations between coefficients $c_n$.

Then there is the Frobenius method, which is also a method for solving ODE, and also in terms of power series.

What is the difference between the power series method and the Frobenius method? Under which conditions would we use either the power series method or the Frobenius method?

Answer 1

The Frobenius method is a generalisation of the power series method. It extends the power series method to include negative and fractional powers. It also allows an extension involving logarithm terms.

So, it uses an Ansatz with

\begin{align*} y(x)&=c_0x^\alpha+c_1x^{\alpha+1}+c_2x^{\alpha+2}+\ldots\\ &=x^\alpha\left(c_0+c_1x^{1}+c_2x^{2}+\ldots\right)\\ &=x^\alpha\sum_{n=0}^{\infty}c_nx^n \end{align*}

and the extension to the power series method is the usage of $x^\alpha$ which is additionally to determine besides the coefficients $a_n$.

The value of $\alpha$ may be positive, negative or a fraction, which considerable extends the range of solutions. A short example together with a discussion of some important aspects is provided in Series Solutions of ODEs - The Frobenius method.