plugin-ch3soln

# plugin-ch3soln - No calculator Closedbook note SHOW ALL...

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No calculator, Closed book & note SHOW ALL YOUR WORK! Spring 2009 Math 245 Please check your discussion section: chapter3Test nllam ol2Pm nlPm (29 Aprit2009) Name: Prob. l. (20 points) Consider the system x'= Ax where the matrix A is given by A=(1, :l ' (4 t) Classiff the fixed point x = 0 of the system as to its stability and type. &r (A-tI)= l,-t t | tA-r)L- 4=o 1,i.I *3eo -r (X-rXltt)=o lq t-rl + ),r%,( 'lrz=-l ( grltk fr h hffi/

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Prob. 2. (25 noints) Consider a system x'= Ax where the matrix I =(o" a'' l fras a pair of \art azz ) complexconjugateeigenvalues 4=o+iot and 2r=o-ico. Thecorrespondingeigenvectorsfor 4 nd lrare found to be of the form v(r) = (a * ib,k)r and v(2) (a - ib,k)r , respectively. Find the real-valued frrnctions u(r) and r(r) such that the general solution of the system can be expressed in the form x(r; C,u(t) +Crw(t) . : q/'€['* ) X =4n* Sc.l .l.lJ ({) (o:'') .'trt?ar)t \ eJ L-o< ) sr /4coSort+ialnnart +ibcotot bs^-t^rt \ e [*^r^* + ;{ s-tnt^rt ) est 1,aroftst - gs,nroa) n i{,rt f as"nut r brslurt) -fqosgot / \ &ir-urt / ar.o!6rt bs". "a t Creo* (a-l^vntat * b col*. \ -fccor"^rt / .-- I -ta frmr^rt'

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## This note was uploaded on 06/01/2009 for the course MATH 245 taught by Professor Alexander during the Spring '07 term at USC.

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plugin-ch3soln - No calculator Closedbook note SHOW ALL...

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