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Questions tagged [frobenius-method]

Use this tag when you want to solve a linear ordinary differential equation with variable coefficients via the Frobenius method.

The Frobenius method or, Method of Frobenius, attributed to the german mathemematican Ferdinand Georg Frobenius , is a generalization of the power series method for finding an infinite series solution for a linear ordinary differential equation (ODE) with variable coefficients. In fact, when an ODE has regular singular points, power series method fails to provide enough linearly independent solutions. In such a case, Frobenius method comes to our aid.

Method:

Assume that $~x_0=0~$ is regular singular point of the differential equation $$P(x)~y''(x)+Q(x)~y'+R(x)~y=0$$

A Frobenius series (generalized Laurent series) of the form $$y=x^r~\sum_{n=0}^{\infty}a_n~x^{n}=\sum_{n=0}^{\infty}a_n~x^{n+r},\qquad a_0\neq 0 $$

can be used to solve the differential equation. The parameter$~r~$ must be chosen so that when the series is substituted into the D.E. the coefficient of the smallest power of $~x~$ is zero. This is called the indicial equation. Next, a recursive equation for the coefficients is obtained by setting the coefficient of$~x^{n+r}~$equal to zero.

Caveat: There are some instances when only one Frobenius solution can be constructed.

References:

https://en.wikipedia.org/wiki/Frobenius_method

http://mathworld.wolfram.com/FrobeniusMethod.html

http://home.iitk.ac.in/~sghorai/TEACHING/MTH203/ode14.pdf

http://www.math.mcgill.ca/gantumur/math315w14/downloads/frobenius.pdf

241 questions
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The Method Of Frobenius

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Can we use Power Series Solution for points other than x=0 to escape Frobenius Solution?

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Is Frobenius' method generally useful?

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Method of Frobenius, constants replaced by power series

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When we solve the equation $z^{2}u''+p(z)zu'+q(z)u=0$ using Frobenius method, we first find zeros of the indicial polynomial $r_1>r_2$ to get a solution $u_1$. In the case that the difference of roots $r_1-r_2$ is an integer, we might need to find a…
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