> | with(plots):with(DEtools): |
Warning, the name changecoords has been redefined
Oscillation general
> | de:=diff(u(t),t$2)+16*diff(u(t),t)+192*u(t)=0; |
> | sol:=dsolve({de,u(0)=1/2,D(u)(0)=0}); |
> | plot(solve(sol,u(t)),t=0..1); |
No dampening
> | de:=diff(u(t),t$2)+192*u(t)=0; |
> | sol:=dsolve({de,u(0)=1/6,D(u)(0)=-1}); |
> | plot(solve(sol,u(t)),t=0..1.5); |
Dampening
> | de:=diff(u(t),t$2)+a*diff(u(t),t)+192*u(t)=0; |
> | sol:=dsolve({de,u(0)=1,D(u)(0)=10}): |
> | sol:=solve(sol,u(t)); |
> | plot(subs(a=1,sol),t=0..2); |
> | plot({subs(a=0,sol),subs(a=5,sol),subs(a=10,sol),subs(a=20,sol)},t=0..1,thickness=3); |
> | plot({subs(a=0,sol),subs(a=27.7,sol),subs(a=60,sol)},t=0..1.6,thickness=3); |
Beats
> | de:=diff(u(t),t$2)+u(t)=1/2*cos(0.8*t); |
> | sol:=dsolve({de,u(0)=0,D(u)(0)=1/4}); |
> | sol:=solve(sol,u(t)); |
> | plot(sol,t=0..50); |
> | plot(sol,t=0..200); |
Resonance
> | de:=diff(u(t),t$2)+u(t)=0.5*cos(t); |
> | sol:=dsolve({de,u(0)=0,D(u)(0)=1/4}); |
> | sol:=solve(sol,u(t)); |
> | plot(sol,t=0..50); |
Forced with Dampening
> | de:=diff(u(t),t$2)+1/8*diff(u(t),t)+u(t)=3*cos(a*t); |
> | sol:=solve(dsolve({de,u(0)=0,D(u)(0)=1/4}),u(t)); |
> | a:=5: |
> | plot(sol,t=0..50); |
> | a:=1/10: |
> | plot(sol,t=0..180); |
> | a:=1: |
> | plot(sol,t=0..100); |
Resonance curve
> | m:=1:omega0:=1: |
> | Delta:=sqrt(m^2*(omega0^2-omega^2)^2+g^2*omega^2); |
> | plot(subs(g=1/2,1/Delta),omega=0.1..2); |
> | display(seq(plot(subs(g=n,1/Delta),omega=0..2),n=[1/16,1/8,1/4,3/8,1/2,1,3])); |
Higher order homogeneous
> | de:=diff(y(t),t$9)-43*diff(y(t),t$6)+496*diff(y(t),t$3)-1728*y(t)=0; |
> | factor(r^9-43*r^6+496*r^3-1728); |
> | solve(r^2+2*r+4=0,r); |
> | solve(r^2+3*r+9=0,r); |
> | dsolve(de); |
Undetermined coefficients
> | de:=diff(y(t),t$4)-2*diff(y(t),t$2)+y(t)=exp(t)+sin(t); |
> | de2:=diff(y(t),t$4)-2*diff(y(t),t$2)+y(t)=0; |
> | dsolve(de2); |
> | dsolve(de); |