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L13 - Series Solutions Let us consider a xed SOLODE P(x)y...

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Series Solutions Let us consider a fixed SOLODE P ( x ) y 00 + Q ( x ) y 0 + R ( x ) y = 0 . Here P, Q, R are assumed to be analytic at a point x 0 (most often they will be poly- nomials), and x 0 is assumed to be an ordi- nary point , i.e., P ( x 0 ) 6 = 0 . Then P ( x ) 6 = 0 in some neighborhood of x 0 , and we may there divide by P ( x ) to obtain y 00 + Q ( x ) P ( x ) | {z } p ( x ) y 0 + R ( x ) P ( x ) | {z } q ( x ) = 0 y 00 + p ( x ) y 0 + q ( x ) y = 0 Our differential equation is thus the usual
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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
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Theorem (Kowalewskaja): Suppose P, Q, R have convergence radii ρ P , ρ Q , ρ R , respec- tively (as power series about x 0 ), and the distance of x 0 to the closest zero of P in the complex plane is δ . Then there exists a FS { y 1 , y 2 } of solutions to our SOLODE above that are analytic about x 0 : y i = X n =0 a i,n ( x - x 0 ) n . ( S i ) The convergence radius of ( S i ) is at least as large as ρ P ρ Q ρ R δ = min( ρ P , ρ Q , ρ R , δ ) . The coefficients a i,n can be found alge- braically by substituting ( S i ) in the SOLODE. 3
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Convergence radius of the solutions ρ P ρ R = C i δ ρ Q min( ρ P , ρ Q , ρ R , δ ) = ρ Q Zeroes of P x 0 δ 4
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Example: Consider the SOLODE L [ y ] def = y 00 + y = 0 . Here P = R = 1 and Q = 0 are polynomials with convergence radii , and δ = as well, inasmuch as the function P ( x ) = 1 has no zero. We expect that all solutions are analytic with convergence radius
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