#Notes on 1.3:

# Test LHS=RHS for u'+ku=kA.
t:='t': u0:='u0': omega:='omega': k:='k':
myANS:=your hand-derived formula for u=uh+up:
LHS:=diff(myANS,t)+k*myANS:
RHS:=k*(38-16*cos(omega*(t-3))):
simplify(expand(LHS-RHS));

#Notes on 1.4

t:='t':omega:='omega':k:='k':
AA:=(t,omega)->38-16*cos(omega*(t-3));
eq:=38-(16*k/(k^2+omega^2))*(k*cos(omega*(t-3))+omega*sin(omega*(t-3))):
uss:=unapply(eq,(t,omega,k));

#Notes on 1.5

t:='t':u0:='u0':k:='k':omega:='omega':
AA:=(t,omega)->38-16*cos(omega*(t-3));
U:=(t,u0,k,omega)-> your answer of u=uh+up from 1.3:
plot({U(t,69,0.3,Pi/12),AA(t,Pi/12)},t=0..48);
plot({uss(t,0.3,Pi/12),AA(t,Pi/12)},t=0..48);

#Notes on 1.6

with(plots):
t:='t':u0:='u0':k:='k':omega:='omega':
U:=(t,u0,k,omega)-> your answer of uh+up from 1.3:
implicitplot(U(t,69,k,Pi/12)=31,t=0..72,k=0.2..0.5);
plot3d({U(t,69,k,Pi/12),31},t=0..72,k=0.2..0.5);