Linear ODEs from an algebraic point of view Inna Scherbak (joint with - PowerPoint PPT Presentation

Linear ODEs from an algebraic point of view Inna Scherbak (joint with Letterio Gatto ) Conference Legacy of Vladimir Arnold Fields Institute, Toronto November 24 – 28, 2014

r We solve the generic order linear ODE         ( ) ( 1) ( 2) r r r r ( ) ( ) ( ) ... ( 1) ( ) 0 u t e u t e u t e u t 1 2 r page 1

r We solve the generic order linear ODE         ( ) ( 1) ( 2) r r r r ( ) ( ) ( ) ... ( 1) ( ) 0 u t e u t e u t e u t 1 2 r 1 ,..., e e indeterminate constant coefficients, r   ( ) [[ ]], [ ,..., ]. u t B t B e e 1 r r r page 1

r We solve the generic order linear ODE         ( ) ( 1) ( 2) r r r r ( ) ( ) ( ) ... ( 1) ( ) 0 u t e u t e u t e u t 1 2 r 1 ,..., e e indeterminate constant coefficients, r   ( ) [[ ]], [ ,..., ]. u t B t B e e 1 r r r   Theorem 1: Let be given by , , h B j j r 1    j . h z     j 2 r r 1 ... ( 1) e z e z e z  j 1 2 r n t      Then ( ) , 0 1, u t h j r   j n j ! n  n j is a fundamental system of solutions. page 1

r We solve the generic order linear ODE         ( ) ( 1) ( 2) r r r r ( ) ( ) ( ) ... ( 1) ( ) 0 u t e u t e u t e u t 1 2 r 1 ,..., e e indeterminate constant coefficients, r   ( ) [[ ]], [ ,..., ]. u t B t B e e 1 r r r   Theorem 1: Let be given by , , h B j j r 1    j . h z     j 2 r r 1 ... ( 1) e z e z e z  j 1 2 r n t      Then ( ) , 0 1, u t h j r   j n j ! n  n j is a fundamental system of solutions.    ( 1 ) 1 ( ) r j u t ( ) ( ) generates this fundamental system: u t u t .    r j 1 r page 1

r We solve the generic order linear ODE         ( ) ( 1) ( 2) r r r r ( ) ( ) ( ) ... ( 1) ( ) 0 u t e u t e u t e u t 1 2 r 1 ,..., e e indeterminate constant coefficients, r   ( ) [[ ]], [ ,..., ]. u t B t B e e 1 r r r   Theorem 1: Let be given by , , h B j j r 1    1 ,..., e e j . the elementary h z r       j 2 r r 1 ... ( 1) 1 , , e z e z e z sym functions in  r j 1 2 r n j  t  , , h the complete sym     Then ( ) , 0 1, u t h j r j     functions: j n j 0 ( 0), h j ! n  j n j     2 1, , ,... h h e h e e is a fundamental system of solutions. 0 1 1 2 1 2    ( 1 ) 1 ( ) r j u t ( ) ( ) generates this fundamental system: u t u t .    r j 1 r page 1

         ( ) ( 1) ( 2) r r r r ( ) ( ) ( ) ... ( 1) ( ) 0 ' X M X u t e u t e u t e u t r 1 2 r  ( ) j ( ) ( ) x t u t j   ( ) x t 0     X     ( )  x t  1 r page 2

         ( ) ( 1) ( 2) r r r r ( ) ( ) ( ) ... ( 1) ( ) 0 ' X M X u t e u t e u t e u t r 1 2 r    ( ) j ( ) ( ) x t u t 0 1 0 0   j   0 0 1 0   ( ) x t     0   M    r X      1 0 0 0  ( )    x t    1 r    1 2 3  r r r     ( 1) ( 1) ( 1) e e e e   1 r r 1 r 2 page 2

         ( ) ( 1) ( 2) r r r r ( ) ( ) ( ) ... ( 1) ( ) 0 ' X M X u t e u t e u t e u t r 1 2 r    ( ) j ( ) ( ) x t u t 0 1 0 0   j   0 0 1 0   ( ) x t     0   M    r X      1 0 0 0  ( )    x t    1 r    1 2 3  r r r     ( 1) ( 1) ( 1) e e e e   1 r r 1 r 2   c 0         ( ) exp , (0) X t M t C C X   r     c  1 r page 2

         ( ) ( 1) ( 2) r r r r ( ) ( ) ( ) ... ( 1) ( ) 0 ' X M X u t e u t e u t e u t r 1 2 r    ( ) j ( ) ( ) x t u t 0 1 0 0   j   0 0 1 0   ( ) x t     0   M    r X      1 0 0 0  ( )    x t    1 r    1 2 3  r r r     ( 1) ( 1) ( 1) e e e e   1 r r 1 r 2   c 0           ( ) j ( ) exp , (0) X t M t C C X ( ) ( ), (0) u t x t u c   r 0 j     c  1 r page 2

         ( ) ( 1) ( 2) r r r r ( ) ( ) ( ) ... ( 1) ( ) 0 ' X M X u t e u t e u t e u t r 1 2 r    ( ) j ( ) ( ) x t u t 0 1 0 0   j   0 0 1 0   ( ) x t     0   M    r X      1 0 0 0  ( )    x t    1 r    1 2 3  r r r     ( 1) ( 1) ( 1) e e e e   1 r r 1 r 2   c 0           ( ) j ( ) exp , (0) X t M t C C X ( ) ( ), (0) u t x t u c   r 0 j     c    1 r exp Remark: M t is the Wronski matrix of the standard r ( ), , ( ) v t v t fundamental system of the solutions to the ODE, ,  0 1 r   ( ) j    (0) , that is (standard initial conditions). v 0 , 1 i j r i ij page 2

  exp Theorem 2: The last column of is our universal fundamental system: M t r    v v v u   0 1 1 1 r r    ' ' '   v v v u       0 1 1 2 r r exp M t  r         ( 1) ( 1) ( 1) r r r   v v v u  0 1 1 0 r   k k  t  t ( ) , ... , ( ) u t h u t h    0 1 1 k r k r ! k ! k    0 1 k k r page 3

The universal and the standard fundamental systems are related as follows,             u v v u   r   1   h h h   1 ( 1)   0 1 e e e 1 0 1 2 r 2 1 r       and         0 1 h h 0 1  1 e r u v v u      ( 1) e           1 r 1 1  1 2 2 1 1 r           0 1 0 0 1                                 v u r u r v 0 0 0 1     0 0 0 1       r r 1 1 page 4

The universal and the standard fundamental systems are related as follows,             u v v u   r   1   h h h   1 ( 1)   0 1 e e e 1 0 1 2 r 2 1 r       and         0 1 h h 0 1  1 e r u v v u      ( 1) e           1 r 1 1  1 2 2 1 1 r           0 1 0 0 1             H E                     v u r u r v 0 0 0 1     0 0 0 1       r r 1 1 HE=I page 4