Differential Equation

DIFFERENTIAL EQUATIONS Gracias al ingeniero Ricardo Lopez, profesor de física de la universidad Tecnológica de Pereira por la solución. Cualquier pregunta puede ser realizada a [email protected] Thanks to proffesor Ricardo Lopez for the solution, he currently belongs to physics department from UNIVERSIDAD TECNOLOGICA DE PEREIRA . Any question can be done to [email protected]

x + 4 x + 13 x = te −t ,    x(0) = 0,  x (0) = 2 m 2 + 4m + 13 = 0,    m = −2 ± 3i xh (t ) = (C1 cos(3t ) + C2 sin(3t ))− t x p (t ) = ( A + Bt )− t x p (t ) = ( B − A)− t − Bt− t x p (t ) = ( A − 2 B )− t + Bt− t ( A − 2 B )− t + Bt− t + 41B − A)− t − 4 Bt− t + 13 A−t + 13B− t = t−t 1 13B + B − 4 B = 1   B = 10 2 1 A − 2 B + 4 B − 4 A + 13 A = 0    ,   10 A = −2 B = −   ∴  A = − 10 50 1 1 x(t ) = (C1 cos(3t ) + C2 sin(3t ))− 2t − − t + t−t 50 10 x(t ) = −2− 3t (C1 cos(3t ) + C2 sin(3t )) + 3− 2t (−C1 sin(3t ) + C2 cos(3t )) + 1 1 = 0, C1 = 50 50 6 1 4 32 x (0) = −2C1 + 3C2 + = 2  ,  C2 = (2 − ) = 50 3 50 50 1 ( cos(3t ) + 32 sin(3t )) − 2t − −t + 5t−t x(t ) = 50 x(0) = C1 −

{

}

1 −t 1 −t 1 −t  +  − t 50 10 10