Homework1_Solutions

# Homework1_Solutions - MA 107 Modeling And Analysis of...

This preview shows pages 1–2. Sign up to view the full content.

MA 107 Modeling And Analysis of Dynamic Systems S06 Professor T-C. Tsao 1. Prove that Item (1) and (2) above are true by direct evaluation. Consider only one repeated characteristic root in (1), i.e. p=2. Item (1): to prove that 11 12 ( ) ... n rt r t hn yt k e k t e ke =+ + + , where r i are roots of the characteristic equation: 1 0 () . . . 0 nn n As s a s as a + + += , is the solution of 1 0 ... 0 n Dy a D y aD y ay ++ + + = , we simply plug the solutions into the differential equation and verify the equality: 1 1 1 1 1 1 1 0 1 0 ,... ... ( ... ) 0 n n n n De re D e r e ra r a r a e == + + = + + = 1 1 1 1 1 1 2 1 1 1 1 1 1 0 1 0 1 1 1 ( 1 ) ()( 1 ) ( 2 ) ,... ... ( ... ) ( ( 1) ... ) n Dte rte D te r rt e r De r n r te r a r a t e n r an r a e −− + = + + + = + + + + + + Note that the ( ) in the first term is zero since r1 is a characteristic root. To prove that second term is also zero we must somehow make use of the fact that r1 is a repeated characteristic root, which means that 1 0 1 2 1 1 1 1 . . . ( ) () ( ... 2( ) ( ) ( ) ( ... 0 n n n sr s s r Ps dA s dP s n s a n s a srP s sr ds ds dA s nr a n r a ds = + + +=− + + = + + + = Thus the ( ) in the second term is also zero.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 8

Homework1_Solutions - MA 107 Modeling And Analysis of...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online