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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