# Frobenius method

Let us consider the linear homogeneous differential equation

$$\sum _{\nu =0}^{n}{k}_{\nu}(x){y}^{(n-\nu )}(x)=\mathrm{\hspace{0.33em}0}$$ |

of order (http://planetmath.org/DifferentialEquation) $n$. If the coefficient functions ${k}_{\nu}(x)$ are continuous^{} and the coefficient ${k}_{0}(x)$ of the highest order derivative (http://planetmath.org/HigherOrderDerivatives) does not vanish on a certain interval (resp. a domain (http://planetmath.org/Domain2) in $\u2102$), then all solutions $y(x)$ are continuous on this interval (resp. ). If all coefficients have the continuous derivatives up to a certain , 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

$${y}^{(n)}(x)+\sum _{\nu =1}^{n}{c}_{\nu}(x){y}^{(n-\nu )}(x)=\mathrm{\hspace{0.33em}0},$$ |

some new coefficients ${c}_{\nu}(x)$ are discontinuous^{} in the singular point. However, if the discontinuity is so, that the products

$$(x-{x}_{0}){c}_{1}(x),{(x-{x}_{0})}^{2}{c}_{2}(x),\mathrm{\dots},{(x-{x}_{0})}^{n}{c}_{n}(x)$$ |

are continuous, and 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 (http://planetmath.org/DifferentialEquation) differential equation. When we use the quotient (http://planetmath.org/Division) forms

$$(x-{x}_{0}){c}_{1}(x):=\frac{p(x)}{r(x)},{(x-{x}_{0})}^{2}{c}_{2}(x):=\frac{q(x)}{r(x)},$$ |

where $r(x)$, $p(x)$ and $q(x)$ are analytic in a neighbourhood of ${x}_{0}$ and $r(x)\ne 0$, our differential equation reads

${(x-{x}_{0})}^{2}r(x){y}^{\prime \prime}(x)+(x-{x}_{0})p(x){y}^{\prime}(x)+q(x)y(x)=\mathrm{\hspace{0.33em}0}.$ | (1) |

Since a 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

${x}^{2}r(x){y}^{\prime \prime}(x)+xp(x){y}^{\prime}(x)+q(x)y(x)=\mathrm{\hspace{0.33em}0},$ | (2) |

where the coefficients have the converging power series^{} expansions

$r(x)={\displaystyle \sum _{n=0}^{\mathrm{\infty}}}{r}_{n}{x}^{n},p(x)={\displaystyle \sum _{n=0}^{\mathrm{\infty}}}{p}_{n}{x}^{n},q(x)={\displaystyle \sum _{n=0}^{\mathrm{\infty}}}{q}_{n}{x}^{n}$ | (3) |

and

$${r}_{0}\ne \mathrm{\hspace{0.33em}0}.$$ |

In the Frobenius method one examines whether the equation (2) allows a series solution of the form

$y(x)={x}^{s}{\displaystyle \sum _{n=0}^{\mathrm{\infty}}}{a}_{n}{x}^{n}={a}_{0}{x}^{s}+{a}_{1}{x}^{s+1}+{a}_{2}{x}^{s+2}+\mathrm{\dots},$ | (4) |

where $s$ is a constant and ${a}_{0}\ne 0$.

Substituting (3) and (4) to the differential equation (2) converts the left hand to

$[{r}_{0}s(s-1)+{p}_{0}s+{q}_{0}]{a}_{0}{x}^{s}+$ | ||

$[[{r}_{0}(s+1)s+{p}_{0}(s+1)+{q}_{0}]{a}_{1}+[{r}_{1}s(s-1)+{p}_{1}s+{q}_{1}]{a}_{0}]{x}^{s+1}+$ | ||

$[[{r}_{0}(s+2)(s+1)+{p}_{0}(s+2)+{q}_{0}]{a}_{2}+[{r}_{1}(s+1)s+{p}_{1}(s+1)+{q}_{1}]{a}_{1}+[{r}_{2}s(s-1)+{p}_{2}s+{q}_{2}]{a}_{0}]{x}^{s+2}+\mathrm{\dots}$ |

Our equation seems clearer when using the notations ${f}_{\nu}(s):={r}_{\nu}s(s-1)+{p}_{\nu}s+{q}_{n}u$:

${f}_{0}(s){a}_{0}{x}^{s}+[{f}_{0}(s+1){a}_{1}+{f}_{1}(s){a}_{0}]{x}^{s+1}+[{f}_{0}(s+2){a}_{2}+{f}_{1}(s+1){a}_{1}+{f}_{2}(s){a}_{0}]{x}^{s+2}+\mathrm{\dots}=\mathrm{\hspace{0.33em}0}$ | (5) |

Thus the condition of satisfying the differential equation by (4) is the infinite system of equations

$\{\begin{array}{cc}{f}_{0}(s){a}_{0}=\mathrm{\hspace{0.33em}0}\hfill & \\ {f}_{0}(s+1){a}_{1}+{f}_{1}(s){a}_{0}=\mathrm{\hspace{0.33em}0}\hfill & \\ {f}_{0}(s+2){a}_{2}+{f}_{1}(s+1){a}_{1}+{f}_{2}(s){a}_{0}=\mathrm{\hspace{0.33em}0}\hfill & \\ \mathrm{\cdots}\hspace{1em}\hspace{1em}\mathrm{\cdots}\hspace{1em}\hspace{1em}\mathrm{\cdots}\hfill & \end{array}$ | (6) |

In the first , since ${a}_{0}\ne 0$, the indicial equation

${f}_{0}(s)\equiv {r}_{0}{s}^{2}+({p}_{0}-{r}_{0})s+{q}_{0}=\mathrm{\hspace{0.33em}0}$ | (7) |

must be satisfied. Because ${r}_{0}\ne 0$, this quadratic equation determines for $s$ two values, which in special case may coincide.

The first of the equations (6) leaves ${a}_{0}\phantom{\rule{veryverythickmathspace}{0ex}}(\ne 0)$ arbitrary. The next linear equations in ${a}_{n}$ allow to solve successively the constants ${a}_{1},{a}_{2},\mathrm{\dots}$ provided that the first coefficients ${f}_{0}(s+1)$, ${f}_{0}(s+2),$ $\mathrm{\dots}$ do not vanish; this is evidently the case when the roots (http://planetmath.org/Equation) of the indicial equation don’t differ by an integer (e.g. when the are complex conjugates^{} or when $s$ is the having greater real part^{}). In any case, one obtains at least for one of the 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 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).

## References

- 1 Pentti Laasonen: Matemaattisia erikoisfunktioita. Handout No. 261. Teknillisen Korkeakoulun Ylioppilaskunta; Otaniemi, Finland (1969).

Title | Frobenius method |

Canonical name | FrobeniusMethod |

Date of creation | 2013-03-22 17:43:49 |

Last modified on | 2013-03-22 17:43:49 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 18 |

Author | pahio (2872) |

Entry type | Topic |

Classification | msc 15A06 |

Classification | msc 34A05 |

Synonym | method of Frobenius |

Related topic | FuchsianSingularity |

Related topic | BesselsEquation |

Related topic | SpecialCasesOfHypergeometricFunction |

Defines | indicial equation |