Read e-book online Global Theory of a Second Order Linear Ordinary Differential PDF

By Yasutaka Sibuya

ISBN-10: 0080871291

ISBN-13: 9780080871295

ISBN-10: 0444109595

ISBN-13: 9780444109590

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Extra resources for Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient

Sample text

1 ) , then, becomes - e-i(&2)'Q(%)y =0 where m Am-h ~ ( 2 =P+ ) z ei m 8hx h=l Therefore, i f we choose 0 so that . SUBDOMINAN T SOLUTIONS 18 the function is ii This means t h a t , i f w e s e t s o l u t i o n of ( 7 . 1 ) . ,w -mk a,) . are s o l u t i o n s of ( 6 . 1 ) . Q,,,(x,a) In particnlar . 1. , am) ; (i! 5) where tively & Y; Ym 1' 3n

2 bhx-h , h=l This i s the same as ( 6 . 1. ,Wl , and 1 m (m: o d d ) (m: even) , . SUBDOMINANT SOLUTIONS 32 This proves ( 6 . l. We can prove ( 6 . 9 ) i n a s i m i l a r manner. 1 can be w r i t t e n as 1 Em(x,a) =-mt2 I n parti-cular bhT Z -1 --1 -h d7 . -1 =x 4~1o +( x-2 ) 1 e q [ -273/2 - a1x2] b,(x,al) and _I 3 _I I i --1 - 1 L ---( --aa ) 2 2 2 8 1 b2(x,y,a2) = x These r e s u l t s agree with those i n Section 8. 16) Y = z eW[J p(T)dTl 7 0 where 1 I p ( x ) =-xTpl[l+ bhx-h] Z . l) t o 2" 2 t a ( x ) 2 I t ['p 1 ( x ) t ' p ( x ) I t i s easy t o see t h a t - P(x) 12 = 0 .

Let p o s i t i v e constants s u f f i c i e n t l y small, while M6 9 P and - Lo p 35 6 p is arbitrary. be dvem. 1 , and p . 2 1. 8). In f a c t we can s e t . L0 = [ s i n ( $ ) l - l (See Fig. 2. 10) 1. > min ( s + wie assumption tha I . KT<+ (See Fig. ) ie = s+zp, I I I LO Flg. 3. 36 Let SUBDOMINANT SOLUTICPJS so = s t Toeie derive T ~ 0> . I sol = be such t h a t min Is+ i0 I 78 . 10) w e %T M sE8 for 6,P 69M6,p a 1 cos 8 2 s i n ( p ) We a l s o have 37 .

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Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient by Yasutaka Sibuya


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Categories: Differential Equations