M257-316Notes_Lecture4

M257-316Notes_Lecture4 - Chapter 4 Lectures 4,5 Ordinary...

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Chapter 4 Lectures 4,5 Ordinary Points and Singular Points Lecture 4 Consider P ( x ) y 0 + Q ( x ) y 0 + R ( x ) y = 0 Homogeneous Eq. (4.1) Divide through by P ( x ): Ly = y 0 + p ( x ) y 0 + q ( x ) y =0 p ( x )= Q/P, R/P (4.2) 4.1 An Ordinary Point: x 0 is said to be an ordinary point of (5.2) if p ( x Q/P and q ( x R/P are analytic at x 0 . i.e. p ( x p 0 + p 1 ( x x 0 )+ ··· = k =0 p k ( x x 0 ) k q ( x q 0 + q 1 ( x x 0 = k =0 q k ( x x 0 ) k Note: (1) If P , Q and R are polynomials then a point x 0 such that P ( x 0 ) 6 =0is an ordinary point. 25
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Lectures 4,5 Ordinary Points and Singular Points (2) If x 0 = 0 is an ordinary point then we assume y = X n =0 c n x n ,y 0 n = X n =1 c n nx n 1 0 n = X n =2 c n n ( n 1) x n 2 0= Ly = X n =2 c n n ( n 1) x n 2 + Ã X n =0 p n x n ! X n =1 nc n x n 1 (4.3) + Ã X n =0 q n x n X n =0 c n x n ! X m =0 © ( m + 2)( m +1) c m +2 + ( p 0 ( m c m +1 + ··· + p m c 1 ) +( q 0 c m + + q m c 0 ) } x m =0 (4.4) yields a non-degenerate recursion for the c m . At an ordinary point x 0 we can obtain two linearly independent solu- tions by power series expansion.
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M257-316Notes_Lecture4 - Chapter 4 Lectures 4,5 Ordinary...

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