restart;EDO:=(1 / omega^2)*diff(y(t),t$2)+(2*xi / omega)*diff(y(t),t) +y(t)=0;

NiM+SSRFRE9HNiIvLCgqJkkmb21lZ2FHRiUhIiMtSSVkaWZmR0kqcHJvdGVjdGVkR0YtNiQtSSJ5R0YlNiNJInRHRiUtSSIkR0YtNiRGMiIiIyIiIkY3KihJI3hpR0YlRjdGKSEiIi1GLDYkRi9GMkY3RjZGL0Y3IiIh

sol_generale:=rhs(dsolve(EDO,y(t)));

NiM+SS1zb2xfZ2VuZXJhbGVHNiIsJiomSSRfQzFHRiUiIiItSSRleHBHNiRJKnByb3RlY3RlZEdGLUkoX3N5c2xpYkdGJTYjLCQqKCwmSSN4aUdGJUYpKiQsJiokRjMiIiNGKSEiIkYpI0YpRjdGOEYpSSZvbWVnYUdGJUYpSSJ0R0YlRilGOEYpRikqJkkkX0MyR0YlRiktRis2IywkKigsJkYzRilGNEYpRilGOkYpRjtGKUY4RilGKQ==

#Afin de pouvoir tracer certaines courbes int\351grales, on se fixe des conditions initialessol_cauchy:=rhs(dsolve({EDO,y(0)=1,D(y)(0)=0},y(t)));

NiM+SStzb2xfY2F1Y2h5RzYiLCYqKCwoKiRJI3hpR0YlIiIjIiIiKiZGKkYsLCZGKUYsISIiRiwjRixGK0YsRi9GLEYsRi5GLy1JJGV4cEc2JEkqcHJvdGVjdGVkR0Y0SShfc3lzbGliR0YlNiMqKCwmRipGLyokRi5GMEYsRixJJm9tZWdhR0YlRixJInRHRiVGLEYsRjAqKCwoRilGLEYvRixGLUYvRixGLkYvLUYyNiMsJCooLCZGKkYsRjlGLEYsRjpGLEY7RixGL0YsRjA=

#Attention MAPLE ne voit pas le cas xi=1. De plus il consid\350re la la racine carr\351 d'une expression qui pet \352tre n\351gative#Regardons quelques exemples num\351riques

pseudo:=subs(xi=0.2,omega=1,y(t));

NiM+SSdwc2V1ZG9HNiIsJiomXiQkIisrKysrXSEjNSQhK0UyaT81RisiIiItSSRleHBHNiRJKnByb3RlY3RlZEdGMkkoX3N5c2xpYkdGJTYjKiZeJCQhIiMhIiIkIityKmV6eipGK0YuSSJ0R0YlRi5GLkYuKiZeJEYpJCIrRTJpPzVGK0YuLUYwNiMqJl4kRjckIStyKmV6eipGK0YuRjxGLkYuRi4=

plot(pseudo,t=0..20);

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

aper:=subs(xi=2,omega=1,y(t));

NiM+SSVhcGVyRzYiLCYqJiwmIiIkIiIiKiRGKSNGKiIiI0YtRiotSSRleHBHNiRJKnByb3RlY3RlZEdGMUkoX3N5c2xpYkdGJTYjKiYsJiEiI0YqRitGKkYqSSJ0R0YlRipGKiNGKiIiJyomLCZGKUYqRitGNkYqLUYvNiMsJComLCZGLUYqRitGKkYqRjdGKiEiIkYqRjg=

plot(aper,t=0..20);

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

crit:=subs(xi=1,omega=1,y(t));

Error, numeric exception: division by zero

#Il faut donc reprendre l'equation diff\351rentiellerestart;omega:=1;

NiM+SSZvbWVnYUc2IiIiIg==

EDO_crit:=(1 / omega^2)*diff(y(t),t$2)+(2 / omega)*diff(y(t),t) +y(t)=0;

NiM+SSlFRE9fY3JpdEc2Ii8sKC1JJWRpZmZHSSpwcm90ZWN0ZWRHRio2JC1JInlHRiU2I0kidEdGJS1JIiRHRio2JEYvIiIjIiIiLUYpNiRGLEYvRjNGLEY0IiIh

crit:=rhs(dsolve({EDO_crit,y(0)=1,D(y)(0)=0},y(t)));

NiM+SSVjcml0RzYiLCYtSSRleHBHNiRJKnByb3RlY3RlZEdGKkkoX3N5c2xpYkdGJTYjLCRJInRHRiUhIiIiIiIqJkYnRjBGLkYwRjA=

plot(crit,t=0..20);

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