#Notes on 1.3:

 # Test LHS=RHS for u'+ku=kA.
 myANS:=your hand-derived formula for u=uh+up:
 LHS:=diff(myANS,t)+k*myANS:
 RHS:=k*(33-13*cos(omega*(t-3))):
 simplify(expand(LHS-RHS));

#Notes on 1.4

 unassign('t','u0','omega','k'):
 AA:=(t,omega)->33-13*cos(omega*(t-3));
 UssEq:=33-(13*k/(k^2+omega^2))*(k*cos(omega*(t-3))+
 omega*sin(omega*(t-3))):
 uss:=unapply(UssEq,(t,omega,k));

#Notes on 1.5

 unassign('t','u0','omega','k'):
 AA:=(t,omega)->33-13*cos(omega*(t-3));
 U:=(t,u0,k,omega)-> your answer of u=uh+up from 1.3:
 plot({U(t,72,0.35,Pi/12),AA(t,Pi/12)},t=0..48);
 plot({uss(t,0.35,Pi/12),AA(t,Pi/12)},t=0..48);

#Notes on 1.6

 with(plots):
 unassign('t','u0','omega','k'):
 U:=(t,u0,k,omega)-> your answer of uh+up from 1.3:
 implicitplot(U(t,72,k,Pi/12)=30,t=0..72,k=0.2..0.4);
 plot3d({U(t,72,k,Pi/12),30},t=0..72,k=0.2..0.4);