# set9 - a39 a38 S e t 9 : S e c o n d O r d e r L i n e a r...

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Unformatted text preview: a39 a38 S e t 9 : S e c o n d O r d e r L i n e a r O D E s- P a r t 1 K y l e A . G a l l i v a n D e p a r t m e n t o f M a t h e m a t i c s F l o r i d a S t a t e U n i v e r s i t y O r d i n a r y D i f f e r e n t i a l E q u a t i o n s F a l l 2 9 1 a39 a38 S e c o n d O r d e r L i n e a r O D E s D e fi n i t i o n 9 . 1 . A n O D E i s s e c o n d o r d e r a n d l i n e a r i f i t c a n b e w r i t t e n P ( t ) y ′ ′ + Q ( t ) y ′ + R ( t ) y = G ( t ) A n a s s o c i a t e d i n i t i a l v a l u e p r o b l e m a l s o s p e c i fi e s y ( t ) = y y ′ ( t ) = y ′ I f P ( t ) negationslash = t h e e x p l i c t f o r m y ′ ′ + p ( t ) y ′ + q ( t ) y = g ( t ) w i l l o f t e n b e u s e d . 2 a39 a38 S e c o n d O r d e r L i n e a r O D E s D e fi n i t i o n 9 . 2 . G i v e n t w o f u n c t i o n s p a n d q , t h e d i f f e r e n t i a l o p e r a t o r L a s s o c i a t e d w i t h t h e O D E t h a t m a p s a t w i c e d i f f e r e n t i a b l e f u n c t i o n o f t o n a n i n t e r v a l α < t < β t o a n o t h e r f u n c t i o n o f t i s g i v e n b y L [ φ ] = φ ′ ′ + p φ ′ + q φ 3 a39 a38 E x i s t e n c e a n d U n i q u e n e s s T h e o r e m 9 . 1 ( T e x t b o o k p a g e 1 4 6 ) . I f p , q , a n d g a r e c o n t i n u o u s f u n c t i o n s o n t h e i n t e r v a l I : α < t < β t h a t c o n t a i n s t t h e n t h e i n i t i a l v a l u e p r o b l e m y ′ ′ + p ( t ) y ′ + q ( t ) y = g ( t ) y ( t ) = y y ′ ( t ) = y ′ h a s a u n i q u e s o l u t i o n t h a t e x i s t t h r o u g h o u t I . 4 a39 a38 S u p e r p o s i t i o n T h e o r e m 9 . 2 ( T e x t b o o k p a g e 1 4 7 ) . I f y 1 a n d y 2 a r e t w o s o l u t i o n s t o t h e h o m o g e n e o u s O D E L [ y ] = y ′ ′ + p ( t ) y ′ + q ( t ) y = t h e n t h e l i n e a r c o m b i n a t i o n c 1 y 1 +...
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## This note was uploaded on 07/21/2011 for the course MAP 2203 taught by Professor Gallian during the Fall '09 term at FSU.

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set9 - a39 a38 S e t 9 : S e c o n d O r d e r L i n e a r...

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