# Frobenius method

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## Frobenius method

[frō′ben·yu̇s ‚meth·əd]
(mathematics)
A method of finding a series solution near a point for a linear homogeneous ordinary differential equation.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Setting (7a) leads to (7b); since [a.sub.0] = 0 would lead to [a.sub.j] = 0 in (6) and Q = 0 in (5), a nontrivial solution requires (7c) leading to the indicial equation (7d) with roots (7e):
This equation can be further expressed as a sum of partial fractions as the roots of the indicial equation [Y.sup.2] + 2[[beta].sup.1/2][e.sup.i[pi]/4]Y/[f.sup.3/2] + i[delta] = 0 are found with the variable [s.sup.1/2] having been converted into y.
Then the method of Frobenius gives the indicial equation for the term [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
Build a program to find the indicial equation and to solve it, in the case of solutions with power series in the neighborhood of a singular regular point of differential equations in the form:
Use the program to calculate the roots of the indicial equation for the differential equation [2.sup.x2]y" - xy' + (1 + x)y = 0
The indicial equation is determined by means of the equation F(r) = r(r - 1) + [p.sub.0]r + [q.sub.0] = 0, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The roots of the indicial equation for the equation 2[x.sup.2]y" - xy' + (1 + x)y = 0 are shown in the figure 12.
Build a program to find "N" coefficients of the power series in the neighborhood of a singular regular point of a differential equation of the form P(x)y" + Q(x)y' + R(x)y = 0, if the roots [r.sub.1] and [r.sub.2] for the indicial equation are known.
According to (4.3) we have F(r)=0, which is the indicial equation already treated in the previous task.
where "ra1" is one of the roots of indicial equation. Then "g" is added to the list "T", using the command:
Notice that the program should be applied twice, once for each root of the indicial equation.
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