Certamen 2 - Matemáticas III (2008) - Forma B.pdf - Departamento de Matem´ aticas UTFSM Certamen 2 MAT-023 Primer Semestre 2008 Tiempo 110 minutos No

Certamen 2 - Matemáticas III (2008) - Forma B.pdf -...

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Departamento de Matem´ aticas, UTFSM Certamen 2, MAT-023, Primer Semestre 2008 Tiempo: 110 minutos. No se puede utilizar calculadora. B 1. Resolver el problema de valor inicial y + 8 y + 20 y = δ π ( t ) + u 5 π ( t ) , y (0) = 0 , y (0) = 1 . Analice el comportamiento a largo plazo de la soluci´ on del problema de valor ini- cial. 25 puntos 2.a) Sean f ( t ) = e - at y g ( t ) = e - 3 t si t 0 y f ( t ) = g ( t ) = 0 si t < 0 donde a es un par´ ametro real. Calcule la convoluci´ on ( f * g )( t ). Sugerencia: Hay un valor del par´ ametro a que debe ser tratado separadamente. b) Sea g ( t ) la funci´ on dada por g ( t ) = 1 si 0 t < 3 0 si 3 t < 7 t - 4 si t 7 . Calcule la trans- formada de Laplace de g ( t ). Sugerencia: Escriba la funci´ on g ( t ) usando funciones Heaviside, es decir u b ( t ) para valores convenientes de b . 25 puntos 3. Sea f ( x ) una funci´ on peri´ odica con peri´ odo 2 π tal que f ( x ) = x π - 1 2 si 0 x π. a) Calcule la serie de Fourier de cosenos de f ( x ). b) Calcule la suma de la serie k =1
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