Math 54 - Quiz 09 Ans, Spring 2006

Math 54 - Quiz 09 Ans, Spring 2006 - . Therefore the...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 54, Quiz 9 Solutions Sections 202/203 GSI Carter 1. Write the word “true” or “false”: Let A be a 3 × 3 matrix with real entries and complex eigenvalue α + βi (that is, α is a real number and β is a nonzero real number). Then A has three distinct eigenvalues. True. The roots of the characteristic polynomial of A must be α + βi , α - βi , and some real number. 2. (a) Solve the following initial value problem: x 0 = ± 2 4 4 2 ² x x (0) = ± 0 1 ² The characteristic polynomial of the matrix is ( λ - 2) 2 - 16 = λ 2 - 4 λ - 12 = ( λ + 2)( λ - 6) , so the eigenvalues are - 2 and 6. The eigenspace for - 2 is the null space of ± - 4 - 4 - 4 - 4 ² which is spanned by (1 , - 1) T . The eigenspace for 6 is the null space of ± 4 - 4 - 4 4 ² which is spanned by (1 , 1) T
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: . Therefore the general solution to the system of differential equations is given by y = c 1 ± 1-1 ² e-2 t + c 2 ± 1 1 ² e 6 t . From the initial condition, we have that ± 1 ² = c 1 ± 1-1 ² + c 2 ± 1 1 ² , and therefore c 1 =-1 2 and c 2 = 1 2 . Therefore the solution to the initial value problem is given by y =-1 2 ± 1-1 ² e-2 t + 1 2 ± 1 1 ² e 6 t . (b) What type of system is the above differential equation? Circle one. i. source ii. sink iii. saddle point iv. spiral in v. spiral out vi. periodic Since the eigenvalues are real and have opposite signs, it is a saddle point. 1...
View Full Document

This note was uploaded on 02/20/2010 for the course AS a taught by Professor 11 during the Spring '10 term at École Normale Supérieure.

Ask a homework question - tutors are online