L07 - REVIEW: SOLODE y 00 + p ( t ) y + q ( t ) y = g ( t )...

Info iconThis preview shows pages 1–5. 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 y 00 + p ( t ) y + q ( t ) y = g ( t ) . Existence and Uniqueness Theorem; The Differential Operator L : f ( t ) 7 f 00 ( t ) + p ( t ) f ( t ) + q ( t ) f ( t ) . Fundamental Systems of Solutions. The Wronskian Definition: The Wronskian or Wronskian Determinant of two functions f 1 ( t ) ,f 2 ( t ) is the determinant W [ f 1 ,f 2 ]( t ) def = f 1 ( t ) f 2 ( t ) f 1 ( t ) f 2 ( t ) def = f 1 ( t ) f 2 ( t )- f 2 ( t ) f 1 ( t ) . 2 Abels Theorem: The Wronskian W ( t ) def = W [ y 1 ,y 2 ]( t ) of two solutions y 1 ,y 2 of any HSOLODE y 00 ( t ) + p ( t ) y ( t ) + q ( t ) y ( t ) = 0 ( HSOL ) satisfies W ( t ) = Ce- R p ( t ) dt . It therefore vanishes never or always. 3 Proof: Write y 1 y 00 2 + p ( t ) y 1 y 2 + q ( t ) y 1 y 2 = 0 y 2 y 00 1 + p ( t ) y 2 y 1 + q ( t ) y 2 y 1 = 0 and subtract: ( y 1 y 00 2- y 2 y 00 1 ) | {z } W + p ( t ) ( y 1 y 2- y 2 y 1 ) | {z } W = 0 implies W + p ( t ) W = 0 . The general solu- tion of this FOLODE is W ( t ) = Ce- R p ( t ) dt . Theorem: Two solutions y 1 ,y 2 of a HSOLODE y 00 ( t ) + p ( t ) y ( t ) + q ( t ) y ( t ) = 0 ( HSOL ) form a fundamental system if and only if their Wronskian W [ y 1 ,y 2 ]( t ) does not van- ish at some, and then at every, point t ....
View Full Document

This note was uploaded on 09/07/2010 for the course PHY 303L taught by Professor Turner during the Spring '08 term at University of Texas at Austin.

Page1 / 15

L07 - REVIEW: SOLODE y 00 + p ( t ) y + q ( t ) y = g ( t )...

This preview shows document pages 1 - 5. Sign up to view the full document.

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