# L08 - Review SOLODE Second Order Linear ODE Existence and...

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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 ( t ) + p ( t ) y ( t ) + q ( t ) y ( t ) = 0 . ( HSOL ) whose Wronskian does not vanish; then we have them all....
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L08 - Review SOLODE Second Order Linear ODE Existence and...

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