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Unformatted text preview: 35 Validating the Method of Frobenius Let us now focus on verifying the claims made in the big theorems of section 34.1: theorem 34.1 on the indicial equation, and theorem 34.2 on solutions about regular singular points. We will begin our work in a rather obvious manner — by applying the basic Frobenius method to a generic differential equation with a regular singular point (after rewriting the equation in a “reduced form”) and then closely looking at the results of these computations. This, along with a theorem on convergence that we’ll discuss, will tell us precisely when the basic method succeeds and why it fails for certain cases. After that, we will derive the alternative solution formulas (formulas (34.3) and (34.4) in theorem 34.2 on page 736) and verify that they truly are valid solutions. Dealing with these later cases will be the challenging part. 35.1 Basic Assumptions and Symbology Throughout this chapter, we are assuming that we have a secondorder linear homogeneous differential equation having a point x on the real line as a regular singular point, and having R as the Frobenius radius of convergence about x . For simplicity, we will further assume x = 0 , keeping in mind that corresponding results can be obtained when the regular singular point is nonzero by using the substitution X = x − x . Also, (after recalling the comments made on page 719 about solutions when x < x ), let us agree that we can restrict ourselves to analyzing the possible solutions on the interval ( , R ) . As noted in lemma 33.2 on page 707, our differential equation can be written as x 2 α( x ) y ′′ + x β( x ) y ′ + γ( x ) y = , (35.1a) where α , β and γ are functions analytic at x = 0 and with α( ) negationslash= 0 . The associated differential equation is then α( x ) y ′′ + β( x ) y ′ + γ( x ) y = . (35.1b) Dividing through by α , we get the corresponding reduced forms for our original differential equation x 2 y ′′ + x P ( x ) y ′ + Q ( x ) y = , (35.2a) and for the associated differential equation y ′′ + P ( x ) y ′ + Q ( x ) y = . (35.2b) 8/14/2011 767 768 Validating the Method of Frobenius In each of these equations, P ( x ) = β( x ) α( x ) and Q ( x ) = γ ( x ) α( x ) . Do observe that, because of the relation between equations (35.2a) and (35.2b), each equation will have the same nonzero singular points as the other. Hence, R is also the radius of analyticity for equation (35.2b). This means (see lemma 32.5 on page 671) that we can express P and Q as power series P ( x ) = ∞ summationdisplay k = p k x k for  x  < R (35.3a) and Q ( x ) = ∞ summationdisplay k = q k x k for  x  < R . (35.3b) For the rest of this chapter, we will be doing computations involving the above p k ’s and q k ’s . Don’t forget this. And don’t forget the relation between P and Q , and the coefficients of the first version of our differential equation. In particular, we might as well note here that p = P ( ) = β( ) α(...
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 Summer '09
 Differential Equations, Equations, Complex differential equation, Frobenius method, Regular singular point, power series method

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