soln_hw8

# soln_hw8 - x 1(0 =-17 and x 2(0 = 22 Section 7.3 16 The...

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Solutions to Homework 8 Section 7.1 3. Set x 1 = u and x 2 = u 0 then x 0 1 = x 2 x 0 2 = - ± 1 - 1 4 t 2 ² x 1 - 1 t x 2 . 4. Set x 1 = u , x 2 = u 0 , x 3 = u 00 , and x 4 = u 000 then x 0 1 = x 2 x 0 2 = x 3 x 0 3 = x 4 x 0 4 = x 1 . 5. Set x 1 = u and x 2 = u 0 then x 0 1 = x 2 x 0 2 = - 0 . 25 x 2 - 4 x 1 + 2 cos(3 t ) . The initial conditions become x 1 (0) = 1 and x 2 (0) = - 2. 22. a. The initial conditions are Q 1 (0) = 25 and Q 2 (0) = 15. Note that the amount of water in each tank remains constant. Therefore Q 0 1 = 1 . 5 - 3 Q 1 30 + 1 . 5 Q 2 20 Q 0 2 = 3 + 3 Q 1 30 - 4 Q 2 20 . b. Set the left hand side to zero and solve for Q 1 and Q 2 . Which gives Q E 1 = 42 and Q E 2 = 36. 1

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c. Substitute in Q 1 = x 1 + 42 and Q 2 = x 2 + 36. Which gives x 0 1 = - 3 x 1 30 + 1 . 5 x 2 20 x 0 2 = 3 x 1 30 - 4 x 2 20 . With initial conditions
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Unformatted text preview: x 1 (0) =-17. and x 2 (0) = 22. Section 7.3 16. The eigenvalues are Î» 1 = 1-2 i and Î» 2 = 1 + 2 i with corresponding eigenvectors x 1 = Â± 1 1 + i Â² x 2 = Â± 1 1-i Â² . 22. The eigenvalues are Î» 1 = 1, Î» 2 = 2 and Î» 3 = 3with corresponding eigenvectors x 1 = 1-1 x 2 = -2 1 x 3 = 1-1 . 2...
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## This note was uploaded on 10/23/2009 for the course MATH 22B taught by Professor Hunter during the Winter '08 term at UC Davis.

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soln_hw8 - x 1(0 =-17 and x 2(0 = 22 Section 7.3 16 The...

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