Graphs for Section 6.4 Homework

Problems 2:

h:=Heaviside(t-Pi)-Heaviside(t-2*Pi);

DE02:=diff(y(t),t$2)+2*diff(y(t),t)+2*y(t)=h;

Y:=dsolve({DE02,y(0)=0,D(y)(0)=0},y(t),method=laplace);

plot([rhs(Y),h],t=0..3*Pi,color=[black, green]);

Problem 4

h:=sin(t)+Heaviside(t-Pi)*sin(t-2*Pi);

DE04:=diff(y(t),t$2)+4*y(t)=h;

Y:=dsolve({DE04,y(0)=0,D(y)(0)=0},y(t),method=laplace);

plot([rhs(Y),h],t=0..4*Pi,color=[black, green]);

Problem 6

h:=Heaviside(t-2);

DE06:=diff(y(t),t$2)+3*diff(y(t),t)+2*y(t)=h;

Y:=dsolve({DE06,y(0)=0,D(y)(0)=1},y(t),method=laplace);

plot([rhs(Y),h],t=0..3*Pi,color=[black, green]);

Problem 9

h:=(t/2)*(1-Heaviside(t-6))+3*Heaviside(t-6);

DE09:=diff(y(t),t$2)+y(t)=h;

Y:=dsolve({DE02,y(0)=0,D(y)(0)=1},y(t),method=laplace);

plot([rhs(Y),h],t=0..13,color=[black, green]);

Problem 16

g:=Heaviside(t-3/2)-Heaviside(t-5/2);

U1:=dsolve({diff(u(t),t$2)+(1/4)*diff(u(t),t)+u(t)=k*g,u(0)=0,D(u)(0)=0},u(t));

with(plots):

animate(plot, [rhs(U1),t=0..4],k=1/2..2); #Think about how you might solve the remaining parts.

Problem 19

h:=1+2*sum('((-1)^k)*Heaviside(t-k*Pi)', 'k'=0..15);

plot(h,t=0..60);

DE19:=diff(y(t),t$2)+y(t)=h;

Y:=dsolve({DE19,y(0)=0,D(y)(0)=0},y(t),method=laplace);

plot(rhs(Y),t=0..60);

Problem 21

h:=1+sum('((-1)^k)*Heaviside(t-k*Pi)', 'k'=0..15);

plot(h,t=0..60);

DE21:=diff(y(t),t$2)+y(t)=h;

Y:=dsolve({DE21,y(0)=0,D(y)(0)=0},y(t),method=laplace);

plot(rhs(Y),t=0..60);