Dear all, I have to study the solution of a retarded ODE

x'(t)=ax(t)+bx(t-t_0)

for t>t_0

The initial data x0:[0,t0]-->R is a function defined in the following way x0(t)=f1(t), for 0<t<t' x0(t)=f2(t), for t'<t<t_0.

My code is the following:

Alpha = Pi/4
solu = NDSolve[ {r'[\[Theta]] ==
    r[\[Theta]] Cot[\[Pi]/3] - r[\[Theta] - 7/3 \[Pi]]/ Sin[\[Pi]/3],
   r[\[Theta] /; 0 <= \[Theta] <= Alpha] ==
    1/(Cos[Alpha]*Sin[Pi/3 + Alpha - \[Theta]]),
   r[\[Theta] /;
      Alpha < \[Theta] <= 2 Pi] == (Tan[Alpha]*Cot[Pi/3] + 1)*
     Exp[(\[Theta] - Alpha)*Cot[Pi/3]],
   r[\[Theta] /;
      2 Pi < \[Theta] <=
       2 Pi + Alpha + Pi/3] == ((Tan[Alpha]*Cot[Pi/3] + 1)*
        Exp[(2 Pi - Alpha)*Cot[Pi/3]] - 1/Cos[Alpha])*
     Exp[Cot[Pi/3]*\[Theta]]}, r, {\[Theta], 0, 4 \[Pi]}]
Plot[Evaluate[{ r[\[Theta]]} /. solu], {\[Theta], 0, 4 \[Pi]}]

The program does not recognize my commands, how can I modify the input of the initial data? Thanks

Attachments:

Schermata 2023-0...png