Lesson 11a -Cauchy Euler

Lesson 11a -Cauchy Euler - ax 2 y'bxy cy 0 y xr ax 2 x r'bx...

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Cauchy Euler Equations C h E l E ti h th f ll i f Cauchy Euler Equations have the following form: 0 ' ' ' 2 cy bxy y ax Now consider the following function: r x y Then the substituting this in will give us: 0 ) ( )' ( ' )' ( 2 r r r x c x bx x ax 0 ) ( 2 c r a b ar 0 ) ( ) ( ) )( 1 )( ( 1 2 2 r r r x c x bxr x r r ax equation indicial the This ac a b a b 4 ) ( ) ( 2 0 ) ( ) ( ) )( 1 )( ( r r r x c x br x r r a This means we have cases to consider! 0 ) )( ( 2 r x c br ar ar 0 ) ( or 0 ) ( 2 r x c r a b ar a r 2 0) (y trivial be ould solution w then the , 0 ) ( If r x
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Solving Second Order Cauchy Euler C 1 “ i t di ti t l l ti d ” Th t i d d t Case 1: “r is two distinct real solutions r 1 and r 2 ”. Then our two independent solutions are: 1 r x y 2 r x y Why? Then the substituting this in will give us: 0 ) ( )' ( ' )' ( 1 1 1 2 r r r x c x bx x ax 0 ) ( ) ( ) )( 1 ( 1 1 1 1 1 1 r r r x c x br x r ar 0 ) ( )' ( ' )' ( 2 2 2 2 r r r x c x bx x ax 0 ) ( ) ( ) )( 1 ( 2 2 2 1 2 2 r r r x c x br x r ar 0 ) )( ) ( ( 1 1 2 1 r x c r a b ar 0 0 0 ) )( ) ( ( 2 2 2 2 r x c r a b ar 0 0 Since we know we can have only 2 independent solutions, our general solution when both are real and distinct is: 2 1 2 1 r r x c x c y
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Solving Second Order Cauchy Euler C 2 “ i l l ti ” Th t i d d t l ti Case 2: “r is one real solutions r”. Then our two independent solutions are:
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