PlanetPhysics/Frobenius Method

Let us consider the linear homogeneous differential equation $$\sum_{\nu=0}^n k_\nu(x) y^{(n-\nu)}(x) = 0$$ of order $$n$$.\, If the coefficient functions $$k_\nu(x)$$ are continuous and the coefficient $$k_0(x)$$ of the highest order derivative does not vanish on a certain interval (resp. a domain in $$\mathbb{C}$$), then all solutions $$y(x)$$ are continuous on this interval (resp. domain).\, If all coefficients have the continuous derivatives up to a certain order, the same concerns the solutions.

If, instead, $$k_0(x)$$ vanishes in a point $$x_0$$, this point is in general a singular point.\, After dividing the differential equation by $$k_0(x)$$ and then getting the form $$y^{(n)}(x)+\sum_{\nu=1}^n c_\nu(x)y^{(n-\nu)}(x) = 0,$$ some new coefficients $$c_\nu(x)$$ are discontinuous in the singular point.\, However, if the discontinuity is restricted so, that the products $$(x-x_0)c_1(x),\quad (x-x_0)^2c_2(x),\quad \ldots,\quad (x-x_0)^nc_n(x)$$ are continuous, and even analytic in $$x_0$$, the point $$x_0$$ is a regular singular point of the differential equation.\\

We introduce the so-called\, Frobenius method \, for finding solution functions in a neighbourhood of the regular singular point $$x_0$$, confining us to the case of a second order differential equation.\, When we use the quotient forms $$(x-x_0)c_1(x) := \frac{p(x)}{r(x)},\quad (x-x_0)^2c_2(x) := \frac{q(x)}{r(x)},$$ where $$r(x)$$, $$p(x)$$ and $$q(x)$$ are analytic in a neighbourhood of $$x_0$$ and\, $$r(x) \neq 0$$,\, our differential equation reads $$\begin{matrix} (x-x_0)^2r(x)y''(x)+(x-x_0)p(x)y'(x)+q(x)y(x) = 0. \end{matrix}$$ Since a simple change\, $$x\!-\!x_0\mapsto x$$\, of variable brings to the case that the singular point is the origin, we may suppose such a starting situation.\, Thus we can study the equation $$\begin{matrix} x^2r(x)y''(x)+xp(x)y'(x)+q(x)y(x) = 0, \end{matrix}$$ where the coefficients have the converging power series expansions $$\begin{matrix} r(x) = \sum_{n=0}^\infty r_nx^n,\quad p(x) = \sum_{n=0}^\infty p_nx^n,\quad q(x) = \sum_{n=0}^\infty q_nx^n \end{matrix}$$ and $$r_0 \neq 0.$$ In the Frobenius method one examines whether the equation (2) allows a series solution of the form $$\begin{matrix} y(x) = x^s\sum_{n=0}^\infty a_nx^n = a_0x^s+a_1x^{s+1}+a_2x^{s+2}+\ldots, \end{matrix}$$ where $$s$$ is a constant and\, $$a_0 \neq 0$$.

Substituting (3) and (4) to the differential equation (2) converts the left hand side to $$\begin{matrix} & [r_0s(s\!-\!1)\!+\!p_0s\!+\!q_0]a_0x^s+\\ & system of equations $$\begin{matrix} \begin{cases} f_0(s)a_0 = 0\\ f_0(s\!+\!1)a_1+f_1(s)a_0 = 0\\ f_0(s\!+\!2)a_2+f_1(s\!+\!1)a_1+f_2(s)a_0 = 0\\ \qquad\cdots\qquad\cdots\qquad\cdots \end{cases} \end{matrix}$$ In the first place, since\, $$a_0 \neq 0$$,\, the indicial equation $$\begin{matrix} f_0(s) \equiv r_0s^2+(p_0-r_0)s+q_0 = 0 \end{matrix}$$ must be satisfied.\, Because\, $$r_0 \neq 0$$,\, this quadratic equation determines for $$s$$ two values, which in special case may coincide.

The first of the equations (6) leaves $$a_0\,(\neq 0)$$ arbitrary.\, The next linear equations in $$a_n$$ allow to solve successively the constants $$a_1,\,a_2,\,\ldots$$ provided that the first coefficients $$f_0(s\!+\!1)$$,\, $$f_0(s\!+\!2),$$\,$$\ldots$$ do not vanish; this is evidently the case when the roots of the indicial equation don't differ by an integer (e.g. when the roots are complex conjugates or when $$s$$ is the root having greater real part).\, In any case, one obtains at least for one of the roots of the indicial equation the definite values of the coefficients $$a_n$$ in the series (4).\, It is not hard to show that then this series converges in a neighbourhood of the origin.

For obtaining the complete solution of the differential equation (2) it suffices to have only one solution $$y_1(x)$$ of the form (4), because another solution $$y_2(x)$$, linearly independent on $$y_1(x)$$, is gotten via mere integrations; then it is possible in the cases\, $$s_1\!-\!s_2 \in\mathbb{Z}$$\, that $$y_2(x)$$ has no expansion of the form (4).