L07 - Review SOLODE, Second Order Linear ODE. Existence and...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Review SOLODE, Second Order Linear ODE. Existence and Uniqueness Theorem. The linear differential operator L [ y ]( t ) def = y 00 ( t ) + p ( t ) y ( t ) + q ( t ) y ( t ) with associated HSOLODE L [ y ] = 0 . ( HSOL ) The Wronskian W ( t ) = W [ y 1 ,y 2 ]( t ) of two solutions { y 1 ,y 2 } of ( HSOL ) is the deter- minant W [ y 1 ,y 2 ]( t ) def = y 1 ( t ) y 2 ( t ) y 1 ( t ) y 2 ( t ) def = y 1 ( t ) y 2 ( t )- y 2 ( t ) y 1 ( t ) . and by Abelss theorem has the form W ( t ) = Ce- R p ( t ) dt . It is either identically zero or never. The pair { y 1 ,y 2 } of solutions of ( HSOL ) is a Fundamental System (FS) if and only if W [ y 1 ,y 2 ]( t ) 6 = 0 for some, and then all, t . In this case every solution y of ( HSOL ) is a linear combination y = c 1 y 1 + c 2 y 2 of y 1 ,y 2 . This yields a substantial Reduction of the problem of finding all solutions of y 00 ( t ) + p ( t ) y ( t ) + q ( t ) y ( t ) = 0 . ( HSOL ) 2 Namely, we need to find only two solu- tions y 1 ,y 2 of y 00...
View Full Document

Page1 / 10

L07 - Review SOLODE, Second Order Linear ODE. Existence and...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online