# Lecture 7 - Solutions of Second Order Linear ODEs The...

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Solutions of Second Order Linear ODEs The Wronskian

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Midterm Details Midterm on WEDNESDAY, in class Covers all sections up to 3.2 EXCEPT 2.4 Bring a Blue Book 1 Page (front and back) of Handwritten notes 50 minutes, 5 questions, 50 points
Solutions of Linear ODEs Remember from Friday: A y 00 + B y 0 + C y = 0

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Solutions of Linear ODEs Remember from Friday: A y 00 + B y 0 + C y = 0 Has general solutions: y 1 = C 1 e r 1 t y 2 = C 2 e r 2 t and
Solutions of Linear ODEs Remember from Friday: A y 00 + B y 0 + C y = 0 Has general solutions: y 1 = C 1 e r 1 t y 2 = C 2 e r 2 t and Which leads to the general form of solution: y = C 1 e r 1 t + C 2 e r 2 t

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Solutions of Linear ODEs Remember from Friday: A y 00 + B y 0 + C y = 0 Has general solutions: y 1 = C 1 e r 1 t y 2 = C 2 e r 2 t and Which leads to the general form of solution: y = C 1 e r 1 t + C 2 e r 2 t y = Or: C 1 e r 1 t C 2 e r 2 t +
Solutions of Linear ODEs Remember from Friday: A y 00 + B y 0 + C y = 0 Has general solutions: y 1 = C 1 e r 1 t y 2 = C 2 e r 2 t and Which leads to the general form of solution: y = C 1 e r 1 t + C 2 e r 2 t y = Or: C 2 e r 2 t + y 1

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Solutions of Linear ODEs Remember from Friday: A y 00 + B y 0 + C y = 0 Has general solutions: y 1 = C 1 e r 1 t y 2 = C 2 e r 2 t and Which leads to the general form of solution: y = C 1 e r 1 t + C 2 e r 2 t y = Or: + y 1 y 2
Solutions of Linear ODEs Remember from Friday: A y 00 + B y 0 + C y = 0 Has general solutions: y 1 = C 1 e r 1 t y 2 = C 2 e r 2 t and Which leads to the general form of solution: y = C 1 e r 1 t + C 2 e r 2 t y = Or: + y 1 y 2

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Solutions of Linear ODEs This holds for general linear ODEs If Then so is y 1 y 2 and are solutions, + y 1 y 2 Why? a 0 ( t ) a 1 ( t ) a 2 ( t ) y 00 y 0 y + + = 0 a 0 ( t ) y 00 + a 1 ( t ) y 0 + a 2 ( t ) y = 0
Solutions of Linear ODEs This holds for general linear ODEs If Then so is y 1 y 2 and are solutions, + y 1 y 2 Why? a 0 ( t ) a 1 ( t ) a 2 ( t ) y 0 y + + = 0 y 00 1 a 0 ( t ) y 00 + a 1 ( t ) y 0 + a 2 ( t ) y = 0

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Solutions of Linear ODEs This holds for general linear ODEs If Then so is y 1 y 2 and are solutions, + y 1 y 2 Why? a 0 ( t ) a 1 ( t ) a 2 ( t ) + + = 0 y 00 1 y 0 1 y a 0 ( t ) y 00 + a 1 ( t ) y 0 + a 2 ( t ) y = 0
Solutions of Linear ODEs This holds for general linear ODEs If Then so is y 1 y 2 and are solutions, + y 1 y 2 Why? a 0 ( t ) a 1 ( t ) a 2 ( t ) + + = 0 y 00 1 y 0 1 y 1 a 0 ( t ) y 00 + a 1 ( t ) y 0 + a 2 ( t ) y = 0

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Solutions of Linear ODEs This holds for general linear ODEs If Then so is y 1 y 2 and are solutions, + y 1 y 2 Why? a 0 ( t ) a 1 ( t ) a 2 ( t ) + + = 0 y 00 1 y 0 1 y 1 a 0 ( t ) y 00 + a 1 ( t ) y 0 + a 2 ( t ) y = 0 a 0 ( t ) a 1 ( t ) a 2 ( t ) + + = 0 y 00 2 y 0 2 y 2
Solutions of Linear ODEs This holds for general linear ODEs If Then so is y 1 y 2 and are solutions, + y 1 y 2 Why? a 0 ( t ) a 1 ( t ) a 2 ( t ) + + = 0 y 00 1 y 0 1 y 1 a 0 ( t ) y 00 + a 1 ( t ) y 0 + a 2 ( t ) y = 0 a 0 ( t ) a 1 ( t ) a 2 ( t ) + + = 0 y 00 2 y 0 2 y 2 So… a 0 ( t ) a 1 ( t ) a 2 ( t ) y 00 y 0 y + + = 0

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Solutions of Linear ODEs This holds for general linear ODEs If Then so is y 1 y 2 and are solutions, + y 1 y 2 Why? a 0 ( t ) a 1 ( t ) a 2 ( t ) + + = 0 y 00 1 y 0 1 y 1 a 0 ( t ) y 00 + a 1 ( t ) y 0 + a 2 ( t ) y = 0 a 0 ( t ) a 1 ( t ) a 2 ( t ) + + = 0 y 00 2 y 0 2 y 2 So… a 0 ( t ) a 1 ( t ) a 2 ( t ) + + ( y 1 + y 2 ) 00 ( y 1 + y 2 ) 0 ( y 1 + y 2 ) a 0 ( t ) a 1 ( t ) a 2 ( t ) y 00 y 0 y + + =
Solutions of Linear ODEs This holds for general linear ODEs If Then so is y 1 y 2 and are solutions, + y 1 y 2 Why?

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• Winter '06
• Mohanty

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