20DF06hw3sol - Math 20D Homework 3 Solutions October 13,...

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Math 20D Homework 3 Solutions October 13, 2006 2.6 8. M y = e x cos y + 3 ,N x = - 3 + e x sin y , so the equation is not exact. 10. M y = 1 /x = N x , so the equation is exact. ψ ( x,y ) = y ln x + 3 x 2 - 2 y y ln x + 3 x 2 - 2 y = C 20. After multiplying by μ we have ( e x sin y - 2 y sin x ) dx + ( e x cos y + 2cos x ) dy = 0 M y = e x cos y - 2sin x = N x , so the new equation is exact. ψ ( x,y ) = e x sin y + 2 y cos x e x sin y + 2 y cos x = C 3.1 6. 4 r 2 - 9 = 0 (2 r - 3)(2 r + 3) = 0 r 1 = 3 / 2 ,r 2 = - 3 / 2 y = c 1 e 3 t/ 2 + c 2 e - 3 t/ 2 11. 6 r 2 - 5 r + 1 = 0 (3 r - 1)(2 r - 1) = 0 r 1 = 1 / 3 ,r 2 = 1 / 2 y = c 1 e t/ 3 + c 2 e t/ 2 y (0) = c 1 + c 2 = 4 ,y 0 (0) = c 1 / 3 + c 2 / 2 = 0 c 1 = 12 ,c 2 = - 8 y = 12 e t/ 3 - 8 e t/ 2 y ( t ) -→ -∞ as t -→ ∞ , since e t/ 2 dominates e y/ 3 . 20. 2 r 2 - 3 r + 1 = 0 (2 r - 1)( r - 1) = 0 r 1 = 1 / 2 ,r 2 = 1 y = c 1 e t/ 2 + c 2 e t y (0) = c 1 + c 2 = 2 ,y 0 (0) = c 1 / 2 + c 2 = 1 / 2 c 1 = 3 ,c 2 = - 1 y = 3 e t/ 2 - e t y ( t ) = 0 where 3
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This note was uploaded on 10/26/2010 for the course MATH 20D 0382332 taught by Professor Edgars during the Spring '10 term at UCSD.

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20DF06hw3sol - Math 20D Homework 3 Solutions October 13,...

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