# N 4 n where n is a root of 1 cosh l cos l 0 1 12 362

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𝜆 n = 𝜇 4 n , where 𝜇 n is a root of 1 + cosh ( 𝜇 L ) cos ( 𝜇 L ) = 0; 𝜆 1 12 . 362 L 4 , 𝜆 2 485 . 52 L 4 𝜙 n ( x ) = sin ( 𝜇 n x ) − sinh ( 𝜇 n x ) sin ( 𝜇 n L ) + sinh ( 𝜇 n L ) cos ( 𝜇 n L ) + cosh ( 𝜇 n L ) ( cos ( 𝜇 n x ) − cosh ( 𝜇 n x )) 25. d. 𝜙 n ( x ) = sin ( 𝜆 n x ) , where 𝜆 n satisfies cos ( 𝜆 n L ) 𝛾 𝜆 n L sin ( 𝜆 n L ) = 0 e. 𝜆 1 1 . 1597 L 2 , 𝜆 2 13 . 276 L 2 Section 11.2, page 543 1. 𝜙 n ( x ) = 2 sin (( n 1 2 ) 𝜋 x ) ; n = 1 , 2 , 2. 𝜙 n ( x ) = 2 ( cos ( n 1 2 ) 𝜋 x ) ; n = 1 , 2 , 3. 𝜙 0 ( x ) = 1 , 𝜙 n ( x ) = 2 cos ( n 𝜋 x ); n = 1 , 2 , 4. 𝜙 n ( x ) = 2 cos ( 𝜆 n x ) (1 + sin 2 𝜆 n ) 1 2 , where 𝜆 n satisfies cos ( 𝜆 n ) 𝜆 n sin ( 𝜆 n ) = 0 5. 𝜙 n ( x ) = 2 e x sin ( n 𝜋 x ); n = 1 , 2 ,

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Boyce 9131 BMAnswersToProblems 2 March 11, 2017 15:55 603 Answers to Problems 603 6. a n = 2 2 (2 n 1) 𝜋 ; n = 1 , 2 , 7. a n = 4 2( 1) n 1 (2 n 1) 2 𝜋 2 ; n = 1 , 2 , 8. a n = 2 2 (2 n 1) 𝜋 (1 − cos ((2 n 1) 𝜋 4)); n = 1 , 2 , 9. a n = 2 2 sin (( n 1 2 )( 𝜋 2 )) ( n 1 2 ) 2 𝜋 2 ; n = 1 , 2 , In Problems 10 through 13, 𝛼 n = ( 1 + sin 2 𝜆 n ) 1 2 and cos 𝜆 n 𝜆 n sin 𝜆 n = 0 . 10. a n = 2 sin 𝜆 n 𝜆 n 𝛼 n ; n = 1 , 2 , 11. a n = 2 ( 2 cos 𝜆 n 1 ) 𝜆 n 𝛼 n ; n = 1 , 2 , 12. a n = 2 ( 1 − cos 𝜆 n ) 𝜆 n 𝛼 n ; n = 1 , 2 , 13. a n = 2 sin ( 𝜆 n 2 ) 𝜆 n 𝛼 n ; n = 1 , 2 , 14. Not self-adjoint 15. Self-adjoint 16. Not self-adjoint 17. Self-adjoint 18. Self-adjoint 21. a. If a 2 = 0 or b 2 = 0, then the corresponding boundary term is missing. 25. a. 𝜆 1 = 𝜋 2 L 2 ; 𝜙 1 ( x ) = sin ( 𝜋 x L ) b. 𝜆 1 (4 . 4934) 2 L 2 ; 𝜙 1 ( x ) = sin ( 𝜆 1 x ) 𝜆 1 x cos ( 𝜆 1 L ) c. 𝜆 1 = (2 𝜋 ) 2 L 2 ; 𝜙 1 ( x ) = 1 − cos (2 𝜋 x L ) 26. 𝜆 1 = 𝜋 2 ( 4 L 2 ) ; 𝜙 1 ( x ) = 1 − cos ( 𝜋 x (2 L )) 27. a. X ′′ ( v D ) X + 𝜆 X = 0 , X (0) = 0 , X ( L ) = 0; T + 𝜆 DT = 0 e. c ( x, t ) = n = 1 a n e 𝜆 n Dt e vx (2 D ) sin ( 𝜇 n x ), where 𝜆 n = 𝜇 2 n + v 2 (4 D 2 ); a n = 4 D 𝜇 2 n L 0 e vx (2 D ) f ( x ) sin ( 𝜇 n x ) dx 2 LD 𝜇 2 n + v sin 2 ( 𝜇 n L ) 28. a. u t + vu x = Du xx , u (0 , t ) = 0 , u x ( L, t ) = 0 , u ( x, 0) = − c 0 b. u ( x, t ) = n = 1 b n e 𝜆 n Dt e vx (2 D ) sin ( 𝜇 n x ), where 𝜆 n = 𝜇 2 n + v 2 (4 D 2 ); b n = 8 c 0 D 2 𝜇 2 n ( e vL (2 D ) ( 2 D 𝜇 n cos ( 𝜇 n L ) + v sin ( 𝜇 n L ) ) 2 D 𝜇 n ) ( v 2 + 4 D 2 𝜇 2 n )(2 LD 𝜇 2 n + v sin 2 ( 𝜇 n L )) Section 11.3, page 553 1. y = 2 n = 1 ( 1) n + 1 sin ( n 𝜋 x ) ( n 2 𝜋 2 2) n 𝜋 2. y = 2 n = 1 ( 1) n + 1 sin (( n 1 2 ) 𝜋 x ) ( ( n 1 2 ) 2 𝜋 2 2 ) ( n 1 2 ) 2 𝜋 2 3. y = − 1 4 4 n = 1 ( cos (2 n 1) 𝜋 x ) ((2 n 1) 2 𝜋 2 2)(2 n 1) 2 𝜋 2 4. y = 2 n = 1 (2 cos 𝜆 n 1) cos ( 𝜆 n x ) 𝜆 n ( 𝜆 n 2) ( 1 + sin 2 𝜆 n ) 5.
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• Spring '16
• Anhaouy
• Districts of Vienna, Boyce, e2t, 3y, = min, + c2 sin x

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