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Frobenius Method


If x_0 is an ordinary point of the ordinary differential equation, expand y in a Taylor series about x_0. Commonly, the expansion point can be taken as x_0=0, resulting in the Maclaurin series

 y=sum_(n=0)^inftya_nx^n.
(1)

Plug y back into the ODE and group the coefficients by power. Now, obtain a recurrence relation for the nth term, and write the series expansion in terms of the a_ns. Expansions for the first few derivatives are

y=sum_(n=0)^(infty)a_nx^n
(2)
y^'=sum_(n=1)^(infty)na_nx^(n-1)
(3)
=sum_(n=0)^(infty)(n+1)a_(n+1)x^n
(4)
y^('')=sum_(n=2)^(infty)n(n-1)a_nx^(n-2)
(5)
=sum_(n=0)^(infty)(n+2)(n+1)a_(n+2)x^n.
(6)

If x_0 is a regular singular point of the ordinary differential equation,

 P(x)y^('')+Q(x)y^'+R(x)y=0,
(7)

solutions may be found by the Frobenius method or by expansion in a Laurent series. In the Frobenius method, assume a solution of the form

 y=x^ksum_(n=0)^inftya_nx^n,
(8)

so that

y=x^ksum_(n=0)^(infty)a_nx^n
(9)
=sum_(n=0)^(infty)a_nx^(n+k)
(10)
y^'=sum_(n=0)^(infty)a_n(n+k)x^(k+n-1)
(11)
y^('')=sum_(n=0)^(infty)a_n(n+k)(n+k-1)x^(k+n-2).
(12)

Now, plug y back into the ODE and group the coefficients by power to obtain a recursion formula for the a_nth term, and then write the series expansion in terms of the a_ns. Equating the a_0 term to 0 will produce the so-called indicial equation, which will give the allowed values of k in the series expansion.

As an example, consider the Bessel differential equation

 x^2(d^2y)/(dx^2)+x(dy)/(dx)+(x^2-m^2)y=0.
(13)

Plugging (◇) into (◇) yields

 sum_(n=0)^infty(k+n)(k+n-1)a_nx^(k+n)+sum_(n=0)^infty(k+n)a_nx^(k+n) 
 +sum_(n=2)^inftya_(n-2)x^(k+n)-m^2sum_(n=0)^inftya_nx^(n+k)=0.
(14)

The indicial equation, obtained by setting n=0, is then

 a_0[k(k-1)+k-m^2]=a_0(k^2-m^2)=0.
(15)

Since a_0 is defined as the first nonzero term, k^2-m^2=0, so k=+/-m. For illustration purposes, ignore k=-m and consider only the case k=m (avoiding the special case m!=1/2), then equation (14) requires that

 a_1(2m+1)=0
(16)

(so a_1=0) and

 [a_nn(2m+n)+a_(n-2)]x^(m+n)=0
(17)

for n=2, 3, ..., so

 a_n=-1/(n(2m+n))a_(n-2)
(18)

for n>1. Plugging back in to (◇), rearranging, and simplifying then gives the series solution that defined the Bessel function of the first kind J_m(x), which is the nonsingular solution to (◇). (Considering the case m=-k proceeds analogously and results in the solution J_(-m)(x)=(-1)^mJ_m(x).)

Fuchs's theorem guarantees that at least one power series solution will be obtained when applying the Frobenius method if the expansion point is an ordinary, or regular, singular point. For a regular singular point, a Laurent series expansion can also be used. Expand y in a Laurent series, letting

 y=c_(-n)x^(-n)+...+c_(-1)x^(-1)+c_0+c_1x+...+c_nx^n+....
(19)

Plug y back into the ODE and group the coefficients by power. Now, obtain a recurrence formula for the c_nth term, and write the Taylor series in terms of the c_ns.


See also

Fuchs's Theorem, Ordinary Differential Equation

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References

Arfken, G. "Series Solutions--Frobenius' Method." §8.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 454-467, 1985.Frobenius. "Ueber die Integration der linearen Differentialgleichungen durch Reihen." J. reine angew. Math. 76, 214-235, 1873.Ince, E. L. Ch. 5 in Ordinary Differential Equations. New York: Dover, 1956.

Referenced on Wolfram|Alpha

Frobenius Method

Cite this as:

Weisstein, Eric W. "Frobenius Method." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FrobeniusMethod.html

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