MAE_305_Midterm_2_solution_key

# MAE_305_Midterm_2_solution_key - MAE 305 Ordinary...

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MAE 305 Ordinary Differential Equations, Fall 2005 Midterm Exam #2 rt e x ξ r r = ( ) 0 = r t t I r A ) sin( 1 ) ( 2 at a a t t h = () 2 2 2 2 2 2 2 1 1 ) ( a s a a as a s s a s H + = + = (a) () ( ) 2 12 / 1 1 3 / 1 2 4 / 1 2 1 1 2 1 ) ( + + + = + + = s s s s s s s Y 2 1 ) ( 2 ) ( ) ( 2 + = s s Y s sY s Y s (b) t t t e e e t y 2 2 12 1 3 1 4 1 ) ( + =

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MAE 305 Ordinary Differential Equations, Fall 2005 PROPER/IMPROPER NODE or SPIRAL POINT (ASYMPT. STABLE) SADDLE POINT ( UNSTABLE ) CENTER or SPIRAL POINT (STABILITY INDETERMINATE) (b) PROPER NODE SPIRAL POINT IMPROPER NODE
MAE 305 Ordinary Differential Equations, Fall 2005

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MAE 305 Ordinary Differential Equations, Fall 2005 0 ' ' ' 0 ' ' = + + = + T T L T X X λ π = + = T T L T T X X ' ' ' ' ' (a) ) ( ) ( ) , ( t T x X t x u = 2 2 2 L n n = L x n x X n sin ) ( = (b) 0 ) ( ) 0 ( = = L X X n=1,2,3,4,5… Note that T n ’(0)=0 + = 1 4 2 sin 1 4 2 cos 1 4 2 exp 2 2 2 n L t n L t n L t T n 0 ' ' ' 2 2 2 = + + n n n T L n T L T + = 1 4 2 sin 1 4 2 cos 1 4 2 exp sin ) , ( 2 2 2 n L t n L t n L t L x n t x u n = + = 1 2 2
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## This note was uploaded on 05/17/2008 for the course MAE 305 taught by Professor Luigimartinelli during the Spring '06 term at Princeton.

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MAE_305_Midterm_2_solution_key - MAE 305 Ordinary...

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