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Asig de transf_y_fourier avanzado
- 1. ASIGNACIN DE EJERCICIOS DE LA UNIDAD III: TRANSFORMADA DE LAPLACE Y SERIES DE FOURIER. VALOR: 10 PUNTOS. 1.- UTILIZAR LA DEFINICION DE TRANSFORMADA DE LAPLACE Y RESOLVER LA SIGUIENTE FUNCION 5 2 F t t 7 5 cos 3 t 3 5 2 t LF t t 7 5 cos 3 t e 1t dt u 3 t 5 2 t t LF t t e 1t dt e 5t 7 dt 5 cos 3 t e 5t dt u 3 u u Resolviendo la 1era Integral t 5 2 1t 5 b t e dt lim b 0 t2 e 1t dt u 3 3 0 u t2 dv e 1t dt 1t 1 1t du 2tdt v e dt e 5 1 1 t2 e 1t dt t2 e 1t e 1t 2tdt 5 5 2 1t t 2 1t 2 1t t e dt e e tdt 5 5 u t dv e 1t dt 1t du dt v e dt 1 1t v e 5 2 1t t2 1t 2 1 1t 1 1t t e dt e t e e dt 5 5 5 5 2 1t t2 1t 2 1t 2 e 1t t e dt e t e C 5 52 52 5 2 1t t2 1t 2 1t 2 1t t e dt e t e e C 5 52 53
- 2. Sustituyendo b t 5 2 1t 5 t2 1t 2 1t 2 1t t e dt lim b 0 e t e e u 3 3 5 52 53 0 t 5 2 1t 5 t2 1t 2 5b 2 5b 2 0 t e dt lim b 0 e e e e u 3 3 5 52 53 53 5 2 1t t 5 2 2 t e dt 3 u 3 3 5 3 52 Resolviendo la Segunda Integral b t 5t b 5t 1 5t e 7 dt lim b 7 e dt 7 lim b e u 0 5 0 t 5t 1 5b 1 0b 1 7 e 7 dt 7 lim b e e 7 u 5 5 5 5 Resolviendo la Tercera Integral t b 5t 5t 5 cos 3 t e dt 5 lim b e cos 3 t dt u 0 b 5t I e cos 3 t dt 0 5t u cos 3 t dv e dt 1 5t du 3 sen 3t v e 5 1 5t 1 5t I cos 3 t e e 3 sen 3t dt 5 5 1 5t 3 I e cos 3 t e 5t sen 3tdt 5 5 u sen 3 t dv e 5t dt 1 5t du 3 cos 3t v e 5 1 5t 3 1 5t 1 5t I e cos 3 t sen 3 t e e 3 cos 3t dt 5 5 5 5 1 5t 3 5t 3 I e cos 3 t 2 e sen 3 t I 5 5 52 3 1 5t 3 5t I 2 I e cos 3 t 2 e sen 3 t C 5 5 5 3 1 5t 1 2 I e 5 cos 3 t 3 sen 3t 5 52 52 3 1 5t I e 5 cos 3 t 3 sen 3t 52 52 1 I 2 5 cos 3 t 3 sen 3t 5 3
- 3. 1 b 5 cos 3 te 1t dt 5 lim b 5 cos 3 t 3 sen 3t 0 52 3 0 1t 5 5e cos 3 tdt 2 lim b b cos 3 b 3 sen 3b 5 0 5 3 1t 52 5e cos 3 tdt 0 52 3 2 7 52 LFt 3 52 5 52 3 2.- UTILIZAR PROPIEDADES Y TABLA PARA DETERMINAR LA TRANSFORMADA DE LAPLACE. ENUNCIE LAS PROPIEDADES ANTES DE RESOLVER. SIMPLIFIQUE LOS RESULTADOS. 7 4t 2 a) F t e ( cos 2 5t 2 cosh2 3t 4t 7 ) 2 3 7 4t F t e cos 2 5t 7e 4t cosh2 3t 14e 4t t 7 3 7 4t L F t L e cos 2 5t 7e 4t cosh2 3t 14e 4t t 7 3 7 L F t L e 4t cos 2 5t 7 L e 4t cosh2 3t 14 L e 4t t 7 3 7 L F t L cos 2 5t ( S 4) 7 L cosh2 3t ( S 4) 14 L t 7 ( S 4) 3 7 S 4 S 4 7! L F t 7L 14 3 ( S 4) 2 2 5 2 ( S 4) 2 2 3 2 ( S 4)8 7 S 4 S 4 70560 L F t 2 7 2 3 ( S 4) 20 ( S 4) 12 ( S 4)8 7( S 4) 7 S ( 4) 70560 LF t 2 2 3( S 8S 36) S 8S 28 ( S 4)8 3 sen3t b) F t t 6 senh 2t 5 2 5 t 18 sen3t Ft t senh 2t 3 5 t 18 sen3t LFt L t senh 2t 3 5 t
- 4. 18 sen3t LFt L t senh2t 3L 5 t 18 d 2 LFt ( 1) 3 L sen3t ds 5 ds s 2 4 0 18 d 2 3 LFt 2 3 2 ds 5 ds s 4 s s 9 18 0 s 2 4 2 2s 3 LFt 3 ds 5 s2 4 s s 2 9 18 0 s2 4 2 2s 1 1 s LFt 2 9 tan 5 s2 4 3 3 s 18 4s 1 1 s LFt 2 3 tan tan 5 s2 4 3 3 72s 1 s LFt 2 3 tan 5 s2 4 2 3 72s 1 s LFt 2 3 3 tan 5 s 8s 16 2 3 3 3 5 c) F t L F" t si F t cos2t 2e 3t t 4 5 3 3 5 Ft cos2t 2e 3t t 4 5 3 3 5 3 5 F0 cos 2 0 2e 3 0 0 2 4 5 4 4 5 F0 4 3 3 5 F't 2 sen 2t 2( 3) e 3t 5t 4 5 3 F't sen 2t 6 e 3t 3t 5 2 3 5 F'0 sen 2(0) 6 e 3( 0 ) 3 0 2 F'0 6
- 5. L F '' t S 2L F t S Y ( 0) Y ' ( 0 ) 3 3 5 3t LF t L cos 2t 2e t 4 5 3 3 5 LF t L cos 2t 2 L e 3t Lt 4 5 3 S 1 3 5! LF t 2 2 4 S 4 S 3 5 S6 3S 2 72 LF t 4 S2 4 S 3 S6 3S 2 72 5 L F '' t S2 S 6 4 S2 4 S 3 S6 4 3S 3 2S 2 72 5S L F '' t 6 4 S2 4 S 3 S4 4 1 3.-Aplicar Tabla, simplificaci坦n y m辿todo correspondiente para determinar L f s F t 3 7 s 5 1 4 5s 5 7 7s 4 4 5 A) L 2 3 3 9 s 2 10s 25 s2 8 18 駕 4 3 s 12 縁器器駕器器 s2 7 c) 4 縁 駕器 b) d) a) 3 3 7 s 5 7 s 4 1 1 4 1 1 5 a) L 1 2 L 2 L 2 3 3 3 3 3 3 s 12 s 4 s 4 4 4 4 3 s 7 1 4 5 1 2 L 2 L 2 3 3 6 3 s 4 s 4 4 4 3 3 7 4 5 4t e cosh 2t e senh 2t 3 6
- 6. 5s 5 7 5s 5 7 b) L 1 3 L1 2 3 9 s2 10s 25 9 s 5 5s 5 7 L1 6 6 9 s 5 9 s 5 5 1 1 7 1 1 L 6 L 6 9 s 5 9 s 5 5 5t t 4 7 5t t 5 1 7 e e e 5t t 4 e 5t t 5 9 5! 9 6! 216 6480 7s 4 7s 4 7 1 s 1 1 1 c) L 1 L1 L L 8s 2 18 9 8 s2 9 2 s2 9 8 s2 4 4 4 3 7 1 s 1 2 1 2 L L 8 9 2 3 9 s2 s2 4 4 7 3 1 3 cosh t senh t 8 2 3 2 2 4 5 7 1 7 d )L 1 4 5 L 2 4 2 2 s2 s 2 7 7 4 35 2 sen t 2 7 2 2 35sen t 7
- 7. 1 4s 7 6s 4 B) L 5 17 1 s2 s s2 s 20 3 4 3 2 2 5 17 5 5 5 17 s2 s s2 s 3 4 3 6 6 4 2 5 32 s 6 9 5 5 4 s 7 4s 7 6 6 2 2 5 32 5 32 s s 6 9 6 9 5 31 5 4 s 4 s 6 3 6 2 2 2 5 32 5 32 s s 6 9 6 3 5 32 4 s 6 31 3 3 2 2 5 2 32 3 32 5 2 32 2 s s 6 3 6 3 5 5 1 4s 7 6 t 32 31 6 t 32 L 4e c os t e sen t 5 17 3 32 3 s2 s 3 4 1 6s 4 L ? 2 1 s s 20 3 2 2 1 1 1 s2 s 20 3 6 6 2 1 719 s 6 36 1 1 1 6 s 4 6 s 6s 4 6 6 6 3 2 2 2 2 1 719 1 719 1 719 1 719 s s s s 6 36 6 36 6 36 6 36
- 8. 1 719 6 s 6 6 6 2 3 2 2 2 1 719 719 1 719 s s 6 6 6 6 1 1 6s 4 6 t 719 18 719 L 6e cos t sen t 1 6 719 6 s2 s 20 3
- 9. 1 s 2 2s 3 C)L s2 2s 2 s 2 2s 5 s 2 25 3 As B Cs D 2 s 2s 2 s 2 2s 5 s2 2s 2 s 2 2s 5 s2 2s 3 As B s 2 2s 5 Cs D s2 2s 2 s2 2s 3 As3 2 As 2 5 As Bs 2 2 Bs 5 B Cs 3 2Cs 2 2Cs Ds 2 2 Ds 2 D s2 2s 3 A B s 3 2A B 2C D s2 5 A 2B 2C 2 D s 5B 2D A B 0 2A B 2C D 1 5A 2 B 2C D 2 5B 2D 3 2 A B 2C D 1 1 5 A 2 B 2C D 2 1 2 A B 2C D 1 5 A 2 B 2C D 2 3A B 1 A B 0 1 3A B 1 A B 0 3A B 1 1 2 A 1; A 2 1 B A; B 2 5B 2D 3 1 3 5 3 5B 2 11 D D 2 2 4 2 A B 2C D 1 1 1 11 17 2 2C 1; 2C 1 2 2 4 4 17 13 13 2C 1 ; 2C C 4 4 8
- 10. 1 1 1 11 s s L1 2 2 2 4 s 2 2s 2 s 2 2s 5 1 1 1 1 1 3 s s 1 1 s 1 2 2 2 2 2 2 2 2 2 s 1 1 s 1 1 s 1 1 1 1 s 2 2 1 1 s 1 3 1 1 L1 2 L 2 L 2 s 2s 2 2 s 1 1 2 s 1 1 1 t 3 t e cost e sent 2 2 1 11 1 11 s s L1 22 4 L1 2 2 4 s 2s 5 s 1 4 1 11 1 13 s 1 1 s 1 L1 2 4 L1 2 4 2 2 s 1 4 s 1 4 1 1 s 1 13 1 1 2 L 2 L 2 2 s 1 4 4 2 s 1 4 1 t 13 t e cos 2t e sen 2t 2 8
- 11. 2 5 4.- Utilizar el teorema de Convoluci坦n y determine: L1 s3 s 2 2 2 5 L1 s3 s2 2 1 1 2 5L 1 s3 s2 2 1 1 t2 L s3 2 1 1 2 L1 L1 s2 2 2 s2 2 1 1 L1 sen 2t s2 2 2 2 1 2 5 t L 2 5 sen 2 t d s3 s2 2 0 2 2 5 t L1 5 2 sen 2 t d 0 s3 s2 2 2 u dv sen 2 t 1 du 2 v cos 2 t 2 2 5 5 L1 2 cos 2 t 2 cos 2t d 3 2 s s 2 2 2 1 1 2 5 5 cos 2 t 2 sen t 2 sen 2 L 2 s3 s2 2 2 2 5 5 1 L1 t 2 cos 2t 2t sen (t ) 2 sen 2 t 3 2 s s 2 2 2
- 12. 5.- Determine el semiperiodo del seno de Fourier para F x 4x ; 0 x 1 realizar el espectro de la funci坦n. 1 t 1 1 bn f ( x)dx 4 x sen(nx) dx T 0 1 0 u 4x dv sen nx dx cos nx du 4dx v n cos nx cos nx bn 4x n n 1 cos nx 4 bn 4x sen (nx) n n2 0 cos n 4 bn 4 sen (n) n n 6.-DESARROLLE LA EXPANSIN Y REALICE EL ESPECTRO DE FOURIER DE LA FUNCIN 1 si 0 x 1 F x T 2 2 x si 1 x 2 1 1 2 A0 dx 2 x dx 2 0 1 2 1 1 x2 A0 x 2x 2 0 2 1 1 12 12 A0 1 0 2.2 2. 1 2 2 2 1 1 1 A0 1 4 2 2 2 2 1 A0 3 2 3 A0 2 1 1 cos nx 2 cos nx An dx 2 x dx 2 0 2 1 2 1 2 1 2 sen nx 2(2 x) sen nx cos nx An 2 n 2 0 n 2 2 1
- 13. 1 2 3 An n n n 1 1 2 bn sen nxdx 2 x sen nxdx 2 0 1 1 1 cos nx cos nx sennx bn 2 x 2 n 2 2 0 1 bn n 2