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Sumario APLICACION DE LA TRANSFORMADA DE LAPLACE

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Este PDF proporciona unas aplicaciones de la Transformada de Laplace, cubriendo sus definiciones básicas, propiedades fundamentales y aplicaciones prácticas en la resolución de ecuaciones diferenciales. Ideal para estudiantes y profesionales que buscan una comprensión rápida y efectiva de este...

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Esta es la demostración de la fórmula:

Usemos las siguientes formulas:

2 1
𝑀(𝑎) = ∫ 𝑒 −𝑎𝑥 sin⁡(𝑏𝑥)(πcoth(𝜋𝑥) − )𝑑𝑥
0 𝑥

1 2𝑥
𝜋 coth(𝜋𝑥) − = ∑ 2
𝑥 𝑥 + 𝑘2
𝑘=1

Hallemos la transformada de laplace:

𝐿 = ∫ 𝑒 −𝑎𝑐 𝑀(𝑎)𝑑𝑎
0
∞ ∞
2 1
𝐿 = ∫ ∫ 𝑒 −𝑎𝑐 𝑒 −𝑎𝑥 sin⁡(𝑏𝑥)(πcoth(𝜋𝑥) − )𝑑𝑥 𝑑𝑎
0 0 𝑥

Integramos con respecto a ‘a’:

1 1
𝐿=∫ (sin⁡(𝑏𝑥)(πcoth(𝜋𝑥) − ))𝑑𝑥
0 𝑥2 + 𝑐 𝑥

Usamos la serie infinita:

1 2𝑥
𝜋 coth(𝜋𝑥) = + ∑ 2
𝑥 𝑥 + 𝑘2
𝑘=1

∞ ∞
2𝑥 sin⁡(𝑏𝑥)𝑑𝑥
𝐿 = ∫ (∑ )
0 𝑥2 +𝑘 2 𝑥2 + 𝑐
𝑘=1

Separamos en fracciones parciales:
∞ ∞
2xsin⁡(𝑏𝑥)𝑑𝑥
𝐿 = ∑∫
(𝑥 2 + 𝑐)(𝑥 2 + 𝑘 2 )
𝑘=1 0
∞ ∞
2 xsin(𝑏𝑥) 1 1
𝐿 = ∑∫ ( 2 − 2 )𝑑𝑥
(𝑘 − 𝑐) (𝑥 + 𝑐) (𝑥 + 𝑘 2 )
2
𝑘=1 0

Seguimos separando:
∞ ∞ ∞ ∞
2 xsin(𝑏𝑥) 1 2 xsin(𝑏𝑥) 1
𝐿 = ∑∫ 2
( 2
) 𝑑𝑥 − ∑ ∫ 𝑑𝑥
0 (𝑘 − 𝑐) (𝑥 + 𝑐) 0 (𝑘 − 𝑐) (𝑥 + 𝑘 2 )
2 2
𝑘=1 𝑘=1

, Aquí, usamos esta fórmula:

xsen⁡(𝑏𝑥)𝑑𝑥 𝜋𝑒 −𝑎𝑏
∫ =
0 𝑥 2 + 𝑎2 2

Aplicamos a la transformada:
∞ ∞ ∞ ∞
2 xsin(𝑏𝑥) 1 2 xsin(𝑏𝑥) 1
𝐿 = ∑∫ (
2 − 𝑐) (𝑥 2 + 𝑐)
) 𝑑𝑥 − ∑ ∫ 2 − 𝑐) (𝑥 2 + 𝑘 2 )
𝑑𝑥
0 (𝑘 0 (𝑘
𝑘=1 𝑘=1
∞ ∞
𝜋𝑒 −√𝑐𝑏 𝜋𝑒 −𝑘𝑏
𝐿 = 2∑ − 2 ∑
2(𝑘 2 − 𝑐) 2(𝑘 2 − 𝑐)
𝑘=1 𝑘=1
∞ ∞
−√𝑐𝑏
1 𝑒 −𝑘𝑏
𝐿 = 𝜋𝑒 ∑ 2 −𝜋 ∑ 2
(𝑘 − 𝑐) (𝑘 − 𝑐)
𝑘=1 𝑘=1

La suma resaltada, lo reemplazamos con esta fórmula:

1 2𝑥
𝜋 coth(𝜋𝑥) − = ∑ 2
𝑥 𝑥 + 𝑘2
𝑘=1

Haciendo

𝑥 = 𝑖 √𝑐

1 2𝑖 √𝑐
𝜋 coth(𝜋𝑖 √𝑐) − =∑
𝑖 √𝑐 −𝑐 + 𝑘 2
𝑘=1

1 𝜋 cot(𝜋𝑖 √𝑐) 1
∑ =− +
(𝑘 2 − 𝑐) 2√𝑐 2𝑐
𝑘=1

Y reemplazamos en ‘L’:

𝜋 cot(𝜋√𝑐) 1 𝑒 −𝑘𝑏
𝐿= 𝜋𝑒 −√𝑐𝑏 (− + )−𝜋∑ 2
2√𝑐 2𝑐 (𝑘 − 𝑐)
𝑘=1

Se viene lo chido, calculemos la transformada inversa de L, para aplicar el teorema de unicidad
para funciones continuas de Laplace, esto dice:

Si dos funciones tienen igual transformada de Laplace, entonces son iguales:

Primero, demonos cuenta de lo siguiente:

𝑒 2𝜋√𝑐𝑖 + 1
cot(𝜋√𝑐) = (𝑖)
(𝑒 2𝜋√𝑐𝑖 − 1)
Por identidad de Euler, un viejo conocido:

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