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ass6 - AM 351 Assignment No 6 Systems of First Order Linear...

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AM 351 Assignment No. 6 Fall 2005 Systems of First Order Linear DEs Due: Tuesday, November 21, 2005, 1:30 p.m. (under my door) 1. Construct the first three iterates u 0 , u 1 and u 2 of Picard’s method of successive substitution applied to the following linear system IVP: " x 0 1 x 0 2 # = " 0 1 - 1 0 # " x 1 x 2 # , " x 1 (0) x 2 (0) # = " 1 0 # . (1) (Work on each component x i separately.) Compare your results with the solution of the IVP. 2. In Assignment No. 5, you determined the fundamental matrix Φ( t, 0) for the linear system defined by x 0 = Ax where A = " - 3 2 2 - 3 # . (2) Of course, we now know that Φ( t, 0) = e t A . Find e t A via an appropriate similarity transformation B = C - 1 AC . 3. Find the constant matrix A corresponding to the linear system x 0 = Ax that has the following general solution: x ( t ) = C 1 e - 2 t 1 1 ! + C 2 e - t 1 - 1 ! . (3) Explain briefly the basis of your solution. 4. Course Notes, Problem Set 3, No. 2. (You can solve for x 1 ( t ) and then use this result to solve for x 2 ( t ).) 5. The force on a charge q due to the presence of an electrostatic field E and a magnetic field B is given by F = q E + q c v × B , where v denotes the velocity of the charged particle and c is the speed of light. (In other words,

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