1.1 notes:
# 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*(35-15*cos(omega*(t-3))):
simplify(expand(LHS-RHS));

1.2 notes:
The answer check uses similar code to 1.1.

1.3 notes:
Modify this plotting example:
a:=0: b:=24: omega:=Pi/12:
f:=t->sin(omega*(t-3))):
plot(f(t),t=a..b);

1.4 notes:
t:='t':u0:='u0':k:='k':omega:='omega':
U:=(t,u0,k,omega)-> your answer of u=uh+up from 1.1:

# Example
U:=(t,u0,k,omega)->u0*exp(-k*t)*cos(omega*t): # Not the 1.1 answer!
u1:=U(t,40,0.3,Pi/12): u2:=U(t,50,0.3,Pi/12):
plot({u1,u2},t=0..72);


1.5 notes:
# define AA, U and uss, then:

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);

1.6 notes:
# Let U:=(t,u0,k,omega)-> your answer of uh+up from 1.1:

with(plots):
t:='t':u0:='u0':k:='k':omega:='omega':
implicitplot(U(t,69,k,Pi/12)=30,t=0..72,k=0.2..0.5);
plot3d({U(t,69,k,Pi/12),30},t=0..72,k=0.2..0.5);