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L07 - REVIEW SOLODE y p(t)y q(t)y = g(t Existence and...

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REVIEW: SOLODE y 00 + p ( t ) y 0 + q ( t ) y = g ( t ) . Existence and Uniqueness Theorem; The Differential Operator L : f ( t ) 7→ f 00 ( t ) + p ( t ) f 0 ( t ) + q ( t ) f ( t ) . Fundamental Systems of Solutions.
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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 0 1 ( t ) f 0 2 ( t ) def = f 1 ( t ) f 0 2 ( t ) - f 2 ( t ) f 0 1 ( t ) . 2
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Abel’s 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 0 ( t ) + q ( t ) y ( t ) = 0 ( HSOL ) satisfies W ( t ) = Ce - R p ( t ) dt . It therefore vanishes never or always. 3
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Proof: Write y 1 y 00 2 + p ( t ) y 1 y 0 2 + q ( t ) y 1 y 2 = 0 y 2 y 00 1 + p ( t ) y 2 y 0 1 + q ( t ) y 2 y 1 = 0 and subtract: ( y 1 y 00 2 - y 2 y 00 1 ) | {z } W 0 + p ( t ) ( y 1 y 0 2 - y 2 y 0 1 ) | {z } W = 0 implies W 0 + 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 0 ( 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 . 4
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Proof of Sufficiency: Suppose then that W [ y 1 , y 2 ]( t 0 ) 6 = 0 . We have to show that ev- ery solution y of ( HSOL ) is a linear com- bination of y 1 , y 2 . Solve c 1 y 1 ( t 0 ) + c 2 y 2 ( t 0 ) = y ( t 0 ) c 1 y 0 1 ( t 0 ) + c 2 y 0 2 ( t 0 ) = y 0 ( t 0 ) and set y ( t ) def = c 1 y 1 (
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