Final10Soln6_10_1

Final10Soln6_10_1 - MA366 Final Last Name: First Name: Show...

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MA366 Final Last Name: First Name: Show all work. A correct answer without supporting work is worth NO credit! (Some calculators can solve differential equations.) There should be no “hard” integrals, unless you mess up somewhere. If this happens, just leave it as an integral and explain how to finish the problem. 1
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2 (1) The following vectors X 1 and Y 1 are eigenvectors for a certain 3 × 3 matrix A corresponding to the eigenvalues 2 - i and - 4 respectively. Find the general solution to the system X 0 = AX in real form . No complex numbers allowed! 5 pts. X 1 = i + 1 i - 2 i , Y 1 = 0 3 1 . Solution: Since e (2 - i ) t = e t cos (2 t ) + i e t sin (2 t ) we see that (1 + i ) e (2 - i ) t = - e t cos (2 t ) - e t sin (2 t ) + i ( e t cos (2 t ) - e t sin (2 t ) ) ie (2 - i ) t = - e t sin (2 t ) + i e t cos (2 t ) - 2 e (2 - i ) t = - 2 e t cos (2 t ) - 2 i e t sin (2 t ) We form matrices from the real and imaginary parts: x 1 ( t ) = - e t cos (2 t ) - e t sin (2 t ) - e t sin (2 t ) - 2 e t cos (2 t ) x 2 ( t ) = e t cos (2 t ) - e t sin (2 t ) e t cos (2 t ) - 2 e t sin (2 t ) The real eigenvector produces the solution x 3 ( t ) = e 4 t 0 3 1 . The general solution is X ( t ) = C 1 x 1 ( t ) + c 2 x 2 ( t ) + c 3 x 3 ( t ) . (2) Given that X 1 , X 2 and X 3 are eigenvectors for the following matrix, find the general solution to X 0 = AX . Hint: To find the eigenvalue, compute AX i . 5 pts. A
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Final10Soln6_10_1 - MA366 Final Last Name: First Name: Show...

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