L13 - Series Solutions Let us consider a fixed SOLODE P x y 00 Q x y R x y = 0 Here P,Q,R are assumed to be analytic at a point x(most often they

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Unformatted text preview: Series Solutions Let us consider a fixed SOLODE P ( x ) y 00 + Q ( x ) y + R ( x ) y = 0 . Here P,Q,R are assumed to be analytic at a point x (most often they will be poly- nomials), and x is assumed to be an ordi- nary point , i.e., P ( x ) 6 = 0 . Then P ( x ) 6 = 0 in some neighborhood of x , and we may there divide by P ( x ) to obtain y 00 + Q ( x ) P ( x ) | {z } p ( x ) y + R ( x ) P ( x ) | {z } q ( x ) = 0 y 00 + p ( x ) y + q ( x ) y = 0 Our differential equation is thus the usual SOLODE. In practice we often do not di- vide by P ( x ) : frequently P,Q,R are poly- nomials and are easier to handle than the analytic functions p = Q/P and q = R/P . 2 Theorem (Kowalewskaja): Suppose P,Q,R have convergence radii P , Q , R , respec- tively (as power series about x ), and the distance of x to the closest zero of P in the complex plane is . Then there exists a FS { y 1 ,y 2 } of solutions to our SOLODE above that are analytic about x : y i = X n =0 a i,n ( x- x ) n . ( S i ) The convergence radius of ( S i ) is at least as large as P Q R = min( P , Q , R , ) . The coefficients a i,n can be found alge- braically by substituting ( S i ) in the SOLODE. 3 Convergence radius of the solutions P R = C i Q min( P , Q , R , ) = Q Zeroes of P x 4 Example: Consider the SOLODE L [ y ] def = y 00 + y = 0 . Here P = R = 1 and Q = 0 are polynomials with convergence radii , and = as well, inasmuch as the function P ( x ) = 1 has no zero. We expect that all solutionshas no zero....
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This note was uploaded on 09/07/2010 for the course PHY 303L taught by Professor Turner during the Spring '08 term at University of Texas at Austin.

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L13 - Series Solutions Let us consider a fixed SOLODE P x y 00 Q x y R x y = 0 Here P,Q,R are assumed to be analytic at a point x(most often they

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