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Unformatted text preview: 34 The Big Theorem on the Frobenius Method, With Applications At this point, you may have a number of questions, including: 1. What do we do when the basic method does not yield the necessary linearly independent pair of solutions? 2. Are there any shortcuts? To properly answer these questions requires a good bit of analysis — some straightforward and some, perhaps, not so straightforward. We will do that in the next chapter. Here, instead, we will present a few theorems summarizing the results of that analysis, and we will see how those results can, in turn, be applied to solve and otherwise gain useful information about solutions to some notable differential equations. By the way, in the following, it does not matter whether we are restricting ourselves to differential equations with rational coefficients, or are considering the more general case in which the coefficients involve fairly arbitrary functions analytic at x . The discussion holds for either case. 34.1 The Big Theorems The Theorems The first theorem simply restates results discussed earlier in lemmas 33.1 and 33.2, and in section 33.5. Theorem 34.1 (the indicial equation and corresponding exponents) Let x be a point on the real line. Then x is a regular singular point for a given second-order, linear homogeneous differential equation if and only if that differential equation can be written as ( x − x ) 2 α( x ) y ′′ + ( x − x )β( x ) y ′ + γ( x ) y = where α , β and γ are all analytic at x with α( x ) negationslash= . Moreover: 8/14/2011 735 736 Frobenius II 1. The indicial equation arising in the method of Frobenius to solve this differential equation is α r ( r − 1 ) + β r + γ = where α = α( x ) , β = β( x ) and γ = γ( x ) . 2. The indicial equation has exactly two solutions r 1 and r 2 (possibly identical). And, if α( x ) , β( x ) and γ( x ) are all real valued, then r 1 and r 2 are either both real valued or are complex conjugates of each other. The next theorem is “the big theorem” of the Frobenius method. It describes generic formulas for solutions about regular singular points, and gives the intervals over which these formulas are valid. Theorem 34.2 (general solutions about regular singular points) Assume x is a regular singular point on the real line for some given second-order homogeneous linear differential equation with real coefficients. Let R be the corresponding Frobenius radius of convergence, and let r 1 and r 2 be the two solutions to the corresponding indicial equation, with r 1 ≥ r 2 . Then, on the intervals ( x , x + R ) and ( x − R , x ) , general solutions to the differential equation are given by y ( x ) = c 1 y 1 ( x ) + c 2 y 2 ( x ) where c 1 and c 2 are arbitrary constants, and y 1 and y 2 are solutions that can be written as follows 1 : 1....
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