Laplace

Soluci´n de EDO Grupo 29,utilizando transformada
o
de laplace
Sebastian Andres Ruiz Peralta 08-09-2014

1

Definicion del sistema

1.1

Definici´n de ecuaciones (ec 1) y (ec 2)
o

(%i1)ec_1: ’diff(y1(t),t,2)= -2*y1(t) + 2*(y2(t)-y1(t));

d2
y1 (t) = 2 (y2 (t) − y1 (t)) − 2 y1 (t)
d t2
(%i2) ec_2: ’diff(y2(t),t,2)= -2*(y2(t)-y1(t)) - 2*y2(t);

(%o1)

(%o2)

1.2

d2
y2(t) = −2 (y2 (t) − y1 (t)) − 2 y2 (t)
d t2

Definici´n condiciones iniciales
o

(%i3)

atvalue (y1(t),t=0,1);

(%o3) 1
(%i4)

atvalue (’diff(y1(t),t),t=0,sqrt(6));
√
(%o4)
6
(%i5)atvalue (y2(t),t=0,1);

(%o5) 1
(%i6)
(%o6)

2

atvalue (’diff(y2(t),t),t=0,-sqrt(6));
√
− 6

Aplicaci´n transformada Laplace
o

(%i7)

lec_1: laplace(ec_1,t,s);
√
(%o7) s2 laplace(y1 (t) , t, s)−s− 6 = 2 (laplace (y2 (t) , t, s) − laplace (y1 (t) , t, s))−2 laplace (y1 (t) , t, s)
(%i8)

lec_2: laplace(ec_2,t,s);
√
(%o8) s2 laplace (y2 (t) , t, s)−s+ 6 = −2 (laplace (y2(t) , t, s) − laplace (y1 (t) , t, s))−2 laplace (y2 (t) , t, s)

2.1

(%i9)

Reordenando y Simplificando la aplicaci´n de la transformada de Laplace
o
a las dos ecuacioneslec_1-(2*(laplace(y2(t),t,s)-laplace(y1(t),t,s))-2*laplace(y1(t),t,s));

(%o9) − 2 (laplace (y2 (t) , t, s) − laplace (y1 (t) , t, s)) + s2 laplace (y1 (t) , t, s) + 2 laplace (y1 (t) , t, s) −
√
s− 6=0

1(%i10) ratsimp(-2*(laplace(y2(t),t,s)-laplace(y1(t),t,s))+s^2*laplace(y1(t),t,s)+
2*laplace(y1(t),t,s)-s-sqrt(6)=0);
√
(%o10) − 2 laplace (y2 (t) , t, s) + s2 + 4 laplace (y1 (t) , t, s) − s − 6 = 0(%i11) lec_2-(-2*(laplace(y2(t),t,s)-laplace(y1(t),t,s))-2*laplace(y2(t),t,s));
(%o11) 2 (laplace (y2 (t) , t, s) − laplace (y1 (t) , t, s)) + s2 laplace (y2 (t) , t, s) + 2 laplace (y2 (t) , t, s) −s +
√
6=0
(%i12) ratsimp(2*(laplace(y2(t),t,s)-laplace(y1(t),t,s))+s^2*laplace(y2(t),t,s)+
2*laplace(y2(t),t,s)-s+sqrt(6)=0);
√
(%o12) s2 + 4 laplace (y2 (t) , t, s) − 2 laplace (y1 (t) , t,...