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nagle_differential_equations_ISM_Part60

# nagle_differential_equations_ISM_Part60 - CHAPTER 10...

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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 ) = 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

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Chapter 10 10. f ( x ) π 2 - 4 π k =0 1 (2 k + 1) 2 cos(2 k + 1) x 12. f ( x ) π 2 6 + 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 + n =1 1 n sin 2 nx 16. f ( x ) n =1 2 [1 - cos ( πn/ 2)] πn sin nx 18. The 2 π -periodic function g ( x ), where g ( x ) = | x | on - π x π 20. The 2 π -periodic function g ( x ), where g
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