SECTION 6.5

Exercises 1, 2, 3, 5, 7, 14, 15, 17, 18, 19,

The first one we'll do by hand using the Laplace transform command.

restart;

with(inttrans):

Eqn1:=diff(y(t),t$2)+2*diff(y(t),t)+2*y(t)=Dirac(t-Pi);

Y1:=laplace(Eqn1,t,s);

Y2:=subs(y(0)=1,D(y)(0)=0,Y1);

Y3:=solve(Y2,laplace(y(t),t,s));

Y4:=invlaplace(Y3,s,t);

plot(Y4,t=0..8);

Eqn2:=diff(y(t),t$2)+4*y(t)=Dirac(t-Pi)-Dirac(t-2*Pi);

Y1:=laplace(Eqn2,t,s);

Y2:=subs(y(0)=0,D(y)(0)=0,Y1);

Y3:=solve(Y2,laplace(y(t),t,s));

Y4:=invlaplace(Y3,s,t);

plot(Y4,t=0..8);

Eqn3:=diff(y(t),t$2)+3*diff(y(t),t)+2*y(t)=Dirac(t-5)+Heaviside(t-10);

Y1:=laplace(Eqn3,t,s);

Y2:=subs(y(0)=0,D(y)(0)=1/2,Y1);

Y3:=solve(Y2,laplace(y(t),t,s));

Y4:=invlaplace(Y3,s,t);

plot(Y4,t=0..20);

Eqn5:=diff(y(t),t$2)+2*diff(y(t),t)+3*y(t)=sin(t)+Dirac(t-3*Pi);

Y1:=laplace(Eqn5,t,s);

Y2:=subs(y(0)=0,D(y)(0)=0,Y1);

Y3:=solve(Y2,laplace(y(t),t,s));

Y4:=invlaplace(Y3,s,t);

plot(Y4,t=0..15);

Eqn7:=diff(y(t),t$2)+y(t)=Dirac(t-2*Pi)*cos(t);

Y1:=laplace(Eqn7,t,s);

Y2:=subs(y(0)=0,D(y)(0)=1,Y1);

Y3:=solve(Y2,laplace(y(t),t,s));

Y4:=invlaplace(Y3,s,t);

plot(Y4,t=0..12);

Eqn14:=diff(y(t),t$2)+g*diff(y(t),t)+y(t)=Dirac(t-1);

Y1:=laplace(Eqn14,t,s);

Y2:=subs(y(0)=0,D(y)(0)=0,Y1);

Y3:=solve(Y2,laplace(y(t),t,s));

Y4:=invlaplace(Y3,s,t); #NOT in a good form, since we have complex solutions

with(plots): #We'll manually enter the function and animate it in g:

A:=sqrt(4-g^2)/2;

H:=(1/A)*exp(-(g/2)*t)*sin(A*t);

animate(H,t=0..4*Pi,g=0..(1/2));

Eqn17:=diff(y(t),t$2)+y(t)=sum(Dirac(t-k*Pi),k=1..20);

Y1:=laplace(Eqn17,t,s);

Y2:=subs(y(0)=0,D(y)(0)=0,Y1);

Y3:=solve(Y2,laplace(y(t),t,s));

Y4:=invlaplace(Y3,s,t);

plot(Y4,t=0..30*Pi);

Eqn18:=diff(y(t),t$2)+y(t)=sum((-1)^(k+1)*Dirac(t-k*Pi),k=1..20);

Y1:=laplace(Eqn18,t,s);

Y2:=subs(y(0)=0,D(y)(0)=0,Y1);

Y3:=solve(Y2,laplace(y(t),t,s));

Y4:=invlaplace(Y3,s,t);

plot(Y4,t=0..30*Pi);

Eqn19:=diff(y(t),t$2)+y(t)=sum(Dirac(t-k*Pi/2),k=1..20);

Y1:=laplace(Eqn19,t,s);

Y2:=subs(y(0)=0,D(y)(0)=0,Y1);

Y3:=solve(Y2,laplace(y(t),t,s));

Y4:=invlaplace(Y3,s,t);

plot(Y4/sin(t),t=0..20*Pi,y=-1..1);