Frobenius method

Let us consider the linear homogeneous differential equation

ν=0nkν(x)y(n-ν)(x)= 0

of order ( n.  If the coefficient functions kν(x) are continuousMathworldPlanetmath and the coefficient k0(x) of the highest order derivative ( does not vanish on a certain interval (resp. a domain ( in ), 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, k0(x) vanishes in a point x0, this point is in general a singular point.  After dividing the differential equationMathworldPlanetmath by k0(x) and then getting the form

y(n)(x)+ν=1ncν(x)y(n-ν)(x)= 0,

some new coefficients cν(x) are discontinuousMathworldPlanetmath in the singular point.  However, if the discontinuity is so, that the products


are continuous, and analytic in x0, the point x0 is a regular singular pointMathworldPlanetmath of the differential equation.

We introduce the so-called  Frobenius methodMathworldPlanetmath  for finding solution functions in a neighbourhood of the regular singular point x0, confining us to the case of a second order ( differential equation.  When we use the quotient ( forms


where r(x), p(x) and q(x) are analytic in a neighbourhood of x0 and  r(x)0,  our differential equation reads

(x-x0)2r(x)y′′(x)+(x-x0)p(x)y(x)+q(x)y(x)= 0. (1)

Since a change  x-x0x  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

x2r(x)y′′(x)+xp(x)y(x)+q(x)y(x)= 0, (2)

where the coefficients have the converging power seriesMathworldPlanetmath expansions

r(x)=n=0rnxn,p(x)=n=0pnxn,q(x)=n=0qnxn (3)


r0 0.

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

y(x)=xsn=0anxn=a0xs+a1xs+1+a2xs+2+, (4)

where s is a constant and  a00.

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


Our equation seems clearer when using the notations  fν(s):=rνs(s-1)+pνs+qnu:

f0(s)a0xs+[f0(s+1)a1+f1(s)a0]xs+1+[f0(s+2)a2+f1(s+1)a1+f2(s)a0]xs+2+= 0 (5)

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

{f0(s)a0= 0f0(s+1)a1+f1(s)a0= 0f0(s+2)a2+f1(s+1)a1+f2(s)a0= 0     (6)

In the first , since  a00,  the indicial equation

f0(s)r0s2+(p0-r0)s+q0= 0 (7)

must be satisfied.  Because  r00,  this quadratic equation determines for s two values, which in special case may coincide.

The first of the equations (6) leaves a0(0) arbitrary.  The next linear equations in an allow to solve successively the constants a1,a2, provided that the first coefficients f0(s+1),  f0(s+2), 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 are complex conjugatesMathworldPlanetmath or when s is the having greater real partMathworldPlanetmath).  In any case, one obtains at least for one of the of the indicial equation the definite values of the coefficients an 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 y1(x) of the form (4), because another solution y2(x), linearly independentMathworldPlanetmath on y1(x), is gotten via mere integrations; then it is possible in the cases  s1-s2  that y2(x) has no expansion of the form (4).


  • 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