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Frob1 - 33 Modied Power Series Solutions and the Basic...

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33 Modified Power Series Solutions and the Basic Method of Frobenius The partial sums of a power series solution about an ordinary point x 0 of a differential equation provide fairly accurate approximations to the equation’s solutions at any point x near x 0 . This is true even if relatively low-order partial sums are used (provided you are just interested in the solutions at points very near x 0 ). However, these power series typically converge slower and slower as x moves away from x 0 towards a singular point, with more and more terms then being needed to obtain reasonably accurate partial sum approximations. As a result, the power series solutions derived in the previous two chapters usually tell us very little about the solutions near singular points. This is unfortunate because, in some applications, the behavior of the solutions near certain singular points can be a rather important issue. Fortunately, in many of those applications, the singular point in question is not that “bad” a singular point, and a modification of the algebraic method discussed in the previous chapters can be used to obtain “modified” power series solutions about these points. That is what we will develop in this and the next chapter. In this chapter, we will only consider second-order homogeneous linear differential equa- tions. One can extend the discussion here to first- and higher-order equations, but the important examples are all second-order. 33.1 Euler Equations and Their Solutions The simplest examples of the sort of equations of interest in this chapter are those discussed back in chapter 19, the Euler equations. Let us quickly review them and take a look at what happens to their solutions about their singular points. Recall that a standard second-order Euler equation is a differential equation that can be written as α 0 x 2 y ′′ + β 0 xy + γ 0 y = 0 where α 0 , β 0 and γ 0 are real constants with α 0 negationslash= 0 . Recall, also, that the basic method for solving such an equation begins with attempting a solution of the form y = x r where r is a 8/14/2011 699
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700 Modified Power Series Solutions and the Basic Method of Frobenius constant to be determined. Plugging y = x r into the differential equation, we get α 0 x 2 bracketleftbig x r bracketrightbig ′′ + β 0 x bracketleftbig x r bracketrightbig + γ 0 bracketleftbig x r bracketrightbig = 0 equal1⇒ α 0 x 2 bracketleftbig r ( r 1 ) x r 2 bracketrightbig + β 0 x bracketleftbig rx r 1 bracketrightbig + γ 0 bracketleftbig x r bracketrightbig = 0 equal1⇒ x r bracketleftbig α 0 r ( r 1 ) + β 0 r + γ 0 bracketrightbig = 0 equal1⇒ α 0 r ( r 1 ) + β 0 r + γ 0 = 0 . The last equation above is the indicial equation, which we typically rewrite as α 0 r 2 + 0 α 0 ) r + γ 0 = 0 , and, from which, we can easily determine the possible values of r using basic algebra.
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