lecture6.mws

>

> with(DEtools):with(plots):NumericEventHandler(invalid_operation=exception):

Warning, the name changecoords has been redefined

Block Spike

> de:=2*diff(y(t),t$2)+diff(y(t),t)+2*y(t)=Heaviside(t-5)-Heaviside(t-20);

de := 2*diff(y(t), `$`(t, 2))+diff(y(t), t)+2*y(t) = Heaviside(t-5)-Heaviside(t-20)

> plot(Heaviside(t-5)-Heaviside(t-20),t=0..30);

[Plot]

> sol:=solve(dsolve({de,y(0)=0,D(y)(0)=0}),y(t)):

> plot(sol,t=0..40);

[Plot]

Increasing input

> g(t):=(Heaviside(t-5)-Heaviside(t-10))*(t-5)/5+Heaviside(t-10);

g(t) := 1/5*(Heaviside(t-5)-Heaviside(t-10))*(t-5)+Heaviside(t-10)

> plot(g(t),t=0..20);

[Plot]

> de:=diff(y(t),t$2)+4*y(t)=g(t);

de := diff(y(t), `$`(t, 2))+4*y(t) = 1/5*(Heaviside(t-5)-Heaviside(t-10))*(t-5)+Heaviside(t-10)

> sol:=solve(dsolve({de,y(0)=0,D(y)(0)=0}),y(t));

sol := 1/20*Heaviside(t-5)*t-1/40*Heaviside(t-5)*sin(2*t-10)-1/4*Heaviside(t-5)-1/20*Heaviside(t-10)*t+1/40*Heaviside(t-10)*sin(2*t-20)+1/2*Heaviside(t-10)sol := 1/20*Heaviside(t-5)*t-1/40*Heaviside(t-5)*sin(2*t-10)-1/4*Heaviside(t-5)-1/20*Heaviside(t-10)*t+1/40*Heaviside(t-10)*sin(2*t-20)+1/2*Heaviside(t-10)

> plot(sol,t=0..16);

[Plot]

Diracspike

> de:=2*diff(y(t),t$2)+diff(y(t),t)+2*y(t)=Dirac(t-5);

de := 2*diff(y(t), `$`(t, 2))+diff(y(t), t)+2*y(t) = Dirac(t-5)

> sol:=solve(dsolve({de,y(0)=0,D(y)(0)=0}),y(t));

sol := 2/15*15^(1/2)*Heaviside(t-5)*exp(5/4-1/4*t)*sin(1/4*15^(1/2)*t-5/4*15^(1/2))

> plot(sol,t=4..25);

[Plot]

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