TRANSFORMADA DE LAPLACE.

TRANSFORMADA DE LAPLACE        

          [pic 1]

 [pic 2]

 [pic 3]

Sea                                                                 [pic 4]

• [pic 5]

La transformada de Laplace de [pic 6]

 [pic 7]

 [pic 8]

Ejemplo:

1. Hallar     cuando  c=cte[pic 9]

 [pic 10]

   [pic 11][pic 12][pic 13]

  s>0 [pic 14]

1.    [pic 15]

1.      [pic 16]

1.     [pic 17]

•    cuando  [pic 18][pic 19]

  [pic 20]

                [pic 21]

                [pic 22]

 [pic 23]

                     [pic 24]

• [pic 25]

                            [pic 26]

 [pic 27]

 [pic 28]

 [pic 29]

•  si a=cte[pic 30]

 [pic 31]

      s>a; s-a>0   o   s>a[pic 32]

•  si w=cte[pic 33]

   [pic 34]

 [pic 35]

• [pic 36]

 [pic 37]

 [pic 38]

1. Demostrar [pic 39]

  [pic 40][pic 41]

 [pic 42]

1. [pic 43]

[pic 44]

1. [pic 45]

=[pic 46][pic 47]

7. Hallar  y       sabemos que    [pic 48][pic 49][pic 50]

 [pic 51]

 [pic 52]

 [pic 53]

Real=    Imaginario= [pic 54][pic 55]

• Hallar [pic 56]

Sabemos que   entonces[pic 57]

[pic 58]

[pic 59]

[pic 60]

[pic 61]

• [pic 62]

 [pic 63]

 [pic 64]

  [pic 65]

•  cuando [pic 66][pic 67]

 [pic 68]

  [pic 69]

•  cuando [pic 70][pic 71]

 [pic 72]

 [pic 73]

 [pic 74]

 [pic 75]

•  cuando [pic 76][pic 77]

   [pic 78][pic 79]

 [pic 80]

  [pic 81]

 [pic 82]

[pic 83]

TRANSFORMADA INVERSA

[pic 84]

         [pic 85]

                                  [pic 86]

                              [pic 87]

                           [pic 88]

         [pic 89]

Ejemplos:

1. [pic 90]
2.      [pic 91]
3.   [pic 92]

         [pic 93]

1.        [pic 94]
2.         [pic 95]
3.  [pic 96]
4.  [pic 97]
5.          [pic 98]
6.            [pic 99]
7.  [pic 100]
8.                     [pic 101]
9.   [pic 102]
10.          [pic 103]
11.          [pic 104]
12.     [pic 105]
13.     [pic 106]
14.          [pic 107]
15.          [pic 108]

 Sabemos que        [pic 109]

 [pic 110]

 [pic 111]

             [pic 112]

1.      [pic 113]

Sabemos que [pic 114]

 [pic 115]

 [pic 116]

 [pic 117]

PROPIEDAD TEOREMA DE TRASLACION

1.            [pic 118][pic 119]

 [pic 120]

1. a=3i[pic 121]

 [pic 122]

1.  [pic 123]

•  [pic 124]

     [pic 125][pic 126]

• [pic 127]
• [pic 128]
•            [pic 129]
•         [pic 130]
•           [pic 131]
• [pic 132]

TRANSFORMADA DE LAPLACE DE LA DERIVADA

[pic 133]

• Demostrar   [pic 134]

  [pic 135]

 [pic 136]

 [pic 137]

 [pic 138]

 [pic 139]

 [pic 140]

• Demostrar   [pic 141]

 [pic 142]

 [pic 143]

  [pic 144]

 [pic 145]

 [pic 146]

 [pic 147]

…

 [pic 148]

1. Usar la transformada de la derivada para hallar ; si sabemos que [pic 149][pic 150]

 [pic 151]

  [pic 152][pic 153]

 [pic 154]

 [pic 155]

1.  Hallar ; si  [pic 156][pic 157]

 [pic 158]

  [pic 159][pic 160]

 [pic 161]

 [pic 162]

 [pic 163]

1. Hallar la transformada de la derivada para hallar [pic 164]

 [pic 165]

   [pic 166][pic 167][pic 168]

 [pic 169]

 [pic 170]

 [pic 171]

 [pic 172]

 [pic 173]

1. Hallar la transformada de la derivada para hallar [pic 174]

   [pic 175][pic 176][pic 177]

 [pic 178]

 [pic 179]

 [pic 180]

 [pic 181]

TRANSFORMADA DE LA INTEGRAL

 [pic 182]

1. Demostrar         [pic 183]

  [pic 184]

 [pic 185]

 [pic 186]

 [pic 187]

 [pic 188]

 [pic 189]

 [pic 190]

 [pic 191]

 [pic 192]

1. Hallar f(t), usando la transformada de la integral:

1. [pic 193]

 [pic 194]

 [pic 195]

 [pic 196]

 [pic 197]

 [pic 198]

 [pic 199]

  [pic 200]

1. [pic 201]

 [pic 202]

 [pic 203]

  [pic 204]

 [pic 205]

 [pic 206]

 [pic 207]

 [pic 208]

1. [pic 209]

 [pic 210]

 [pic 211]

 [pic 212]

 [pic 213]

 [pic 214]

 [pic 215]

APLICACIONES (ECUACIONES DIFERENCIALES)

1. Resolver: [pic 216]

 [pic 217]

 [pic 218]

 [pic 219]

 [pic 220]

 [pic 221]

 [pic 222]

 [pic 223]

 [pic 224]

1. [pic 225]

 [pic 226]

  [pic 227]

 [pic 228]

 [pic 229]

 [pic 230]

 [pic 231]

 [pic 232]

 [pic 233]

 [pic 234]

 [pic 235]

 [pic 236]

DERIVADA DE LA TRANSFORMADA

 [pic 237]

Donde: [pic 238]

Para [pic 239]

  [pic 240]

F(s)= [pic 241]

  [pic 242]

 [pic 243]

Para [pic 244]

 [pic 245]

  [pic 246]

 [pic 247]

Ejemplos

1. [pic 248]

 [pic 249]

 [pic 250]

 [pic 251]

 [pic 252]

 [pic 253]

1. [pic 254]

 [pic 255]

 [pic 256]

 [pic 257]

 [pic 258]

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