nagle_differential_equations_ISM_Part60

nagle_differential_equations_ISM_Part60 - CHAPTER 10:...

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Unformatted text preview: CHAPTER 10: Partial Differential Equations EXERCISES 10.2: Method of Separation of Variables 2. y = ( e 10- 1) e x + (1- e 2 ) e 5 x e 10- e 2 4. y = 2 sin 3 x 6. No solution 8. y = e x- 1 + xe x- 1 10. λ n = (2 n- 1) 2 4 and y n = c n cos 2 n- 1 2 x , where n = 1 , 2 , 3 ,... and c n ’s are arbitrary 12. λ n = 4 n 2 and y n = c n cos(2 nx ), where n = 0 , 1 , 2 ,... and c n ’s are arbitrary 14. λ n = n 2 + 1 and y n = c n e x sin( nx ), where n = 1 , 2 , 3 ,... and c n ’s are arbitrary 16. u ( x,t ) = e- 27 t sin 3 x + 5 e- 147 t sin 7 x- 2 e- 507 t sin 13 x 18. u ( x,t ) = e- 48 t sin 4 x + 3 e- 108 t sin 6 x- e- 300 t sin 10 x 20. u ( x,t ) =- 2 9 sin 9 t sin 3 x + 3 7 sin 21 t sin 7 x- 1 30 sin 30 t sin 10 x 22. u ( x,t ) = cos 3 t sin x- cos 6 t sin 2 x +cos 9 t sin 3 x + 2 3 sin 9 t sin 3 x- 7 15 sin 15 t sin 5 x 24. u ( x,t ) = ∞ X n =1 1 n 2 cos 4 nt + (- 1) n +1 4 n 2 sin 4 nt sin nx EXERCISES 10.3: Fourier Series 2. Even 4. Neither 6. Odd 291 Chapter 10 10. f ( x ) ∼ π 2- 4 π ∞ X k =0 1 (2 k + 1) 2 cos(2 k + 1) x 12. f ( x ) ∼ π 2 6 + ∞ X n =1 2(- 1) n n 2 cos nx + (- 1) n (2- n 2 π 2 )- 2 πn 3 sin nx 14. f ( x ) ∼ π 2 + ∞ X n =1 1 n sin 2 nx 16. f ( x ) ∼ ∞ X n =1 2 [1- cos (...
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This note was uploaded on 02/12/2011 for the course MA 221 taught by Professor Mazmani during the Spring '08 term at Stevens.

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nagle_differential_equations_ISM_Part60 - CHAPTER 10:...

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