22BsolnsHW8

22BsolnsHW8 - Math 22B Solutions Homework 8 Spring 2008...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 22B Solutions Homework 8 Spring 2008 Section 7.5 2. Solution Let x = ζert . Substitute into the ODE to get: 1−r −2 3 −4 − r ζ1 ζ2 0 0 = For a nonzero solution, we must have det(A − rI ) = r2 +3r +2 = 0, which has roots r1 = −1 and r2 = −2. For r = −1, we get ζ1 = ζ2 . A corresponding eigenvector is v1 = (1, 1)T . For r = −2, we get 3ζ1 = 2ζ2 . A corresponding eigenvector is v2 = (2, 3)T . So x = c1 (1, 1)T e−t + c2 (2, 3)T e−2t . Also see figure 1 for the graph. 24. See figure 2 Section 7.6 13. Solution (a) det(A − rI ) = (α − r)2 + 1 so r2 − 2rα + α2 + 1 = 0 gives us r = α ± i. (b) The equilibrium is when α = 0. (c) See figure 3. 28. Solution (a) x1 = u, x2 = u ⇒ x1 = x2 and x2 = u . So x2 = −k x. m1 (b) det(A − rI ) = r2 + k m = 0. So r = ±i −k u m k . m (c) Origin is the center. See figure 4. (d) The general solution is u(t) = c1 cos k t + c2 sin m k t m So the solution to the system is x = c1 ( m sin k k , cos m kT t) + c2 ( m 1 m cos k k , − sin m kT t) m = The natural frequency is Im (r1,2 ). Section 7.7 3. Solution The eigenvalues and eigenvectors were found in Prob. 3 of section 7.5. The general soln of the system is x = c1 (1, 1)T et + c2 (1, 3)T e−t . Given the initial conditions x(0) = e, we solve the equations c1 + c2 = 1 3 and c1 + 3c2 = 0, to obtain c1 = 2 , c2 = −1 . The corresponding solution 2 3t 1 −t 3 t 3 −t T is x = ( 2 e − 2 e , 2 e − 2 e ) . Given the initial conditions x(0) = e(2) , we solve the equations c1 + c2 = 0 and c1 + 3c2 = 1 to obtain c1 = −1 and 2 1 c2 = 1 . The corresponding equation is x = ( −1 et + 2 e−t , −1 et + 3 e−t )T . The 2 2 2 2 fundamental matrix easily follows. 2 ...
View Full Document

Page1 / 2

22BsolnsHW8 - Math 22B Solutions Homework 8 Spring 2008...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online