Math 2280-2 Free damped motions
The kind of motion depends on the sign of p^2 -w0^2. See the textbook pp190-192.
Underdamped free motion
with(DEtools):
p:=0.5; w0:=4;
deqtn:= diff(x(t),t,t) + 2*p*diff(x(t),t) + w0^2*x(t) = 0;
# note the syntax for double derivatives
dsolve(deqtn);
DEplot(deqtn,x(t),t=0..5,[[x(0)=1,D(x)(0)=1]],stepsize=1/100,linecolor=[blue]);
# note the syntax of the initial condition
Critically damped free motion
This case
p:=4; w0:=4;
deqtn:= diff(x(t),t,t) + 2*p*diff(x(t),t) + w0^2*x(t) = 0;
dsolve(deqtn);
DEplot(deqtn,x(t),t=0..1,[[x(0)=1,D(x)(0)=1],[x(0)=1,D(x)(0)=0],[x(0)=1,D(x)(0)=-1]],stepsize=1/100,linecolor=[blue,green,red]);
Damped free motion
p:=4; w0:=1;
deqtn:= diff(x(t),t,t) + 2*p*diff(x(t),t) + w0^2*x(t) = 0;
dsolve(deqtn);
DEplot(deqtn,x(t),t=0..10,[[x(0)=1,D(x)(0)=1],[x(0)=1,D(x)(0)=0],[x(0)=1,D(x)(0)=-1]],stepsize=1/200,linecolor=[blue,green,red]);