NO!!!

YOURSsolution is NOT a solution of ODE, even if calculated by hand, and it does NOT fulfill YOURSsolution[ 0 ] == 0.

In[150]:= YOURSsolution /. t -> 0
% // FullSimplify

Out[150]= 1/2 B r (1/(R - I (w - w0)) + 1/(R + I (w - w0)))    
Out[151]= (B r R)/(R^2 + (w - w0)^2)

You may drop the condition y[ 0 ] == 0 in your ODE to get

ysol1 = y /. DSolve[{y'[t] == (I*w0 - R)*y[t] + B*r/2*(Exp[I*w*t] + Exp[-I*w*t])}, y, t][[1, 1]]

Then drop, as you wanted to, C[1]

ysol2 = ysol1 /. C[1] -> 0

But

In[171]:= ysol2[0]
Out[171]= 1/2 B r (1/(R + I (w - w0)) + 1/(R - I (w + w0)))

That is NOT what you wanted.

ysol2 is a solution of ODE

In[172]:= 
ysol2'[t] == (I*w0 - R)*ysol2[t] +    B*r/2*(Exp[I*w*t] + Exp[-I*w*t]) // FullSimplify

Out[172]= True

ysol2 looks very similar to YOURSsolution (but remember: it does not fulfill your initial condition)

In[181]:= (B r)/2 (2/(B r) ysol2[t] // Simplify // Expand)

Out[181]= 1/2 B r (E^(I t w)/(R + I (w - w0)) + E^(-I t w)/(  R - I (w + w0)))

but it is not the same

In[173]:= ysol2[t] - YOURSsolution // FullSimplify
Out[173]= (I B E^(-I t w) r w0)/((R - I w)^2 + w0^2)

Look at the signs in the w / w0 terms , I have the suspicion that there is a typo in that dissertation.

Thanks for your return again. My problem is that I want to get the solution form of YOURSsolution when I solve the equation with Mathematica. I have known that ysol is generally a solution. I think YOURSsolution is a special solution. I have solved the problem by the code that I posted. thanks~

Thanks for your complete return. if I use the code below ,I can got the answer I expected.

In[16]:=  ODE = {y'[t] == (I*w0 - R)*y[t] + B/2*r*(Exp[I*w*t] + Exp[-I*w*t])};
    sol = DSolve[ODE, y, t] // FullSimplify
Out[17]:={{y -> Function[{t}, 
    1/2 B E^(t (-R + I w0) + t (R - I (w + w0)))
       r (E^(2 I t w)/(R + I (w - w0)) + 1/(R - I (w + w0))) + 
     E^(t (-R + I w0)) C[1]]}}                                                                                                        
In[18]:= Simplify[1/2 B E^(t (-R + I w0) + t (R - I (w + w0)))
                 r (E^(2 I t w)/(R + I (w - w0)) + 1/(R - I (w + w0)))]                                                
 Out[19]:= 1/2 B E^(-I t w) r (E^(2 I t w)/(R + I (w - w0)) + 1/(R - I (w + w0))) 

but I still confuse why my expected answer is wrong when I instead the eqution with it.

ODE = {y'[t] == (I*w0 - R)*y[t] + B/2*r*(Exp[I*w*t] + Exp[-I*w*t])};
sol = DSolve[ODE, y, t] // FullSimplifyere                                                                    
  YOURSsolution = {{y -> 
 Function[{t}, (B*r)/
    2*(Exp[I w t]/(R + I*(w - w0)) + 
     Exp[-I w t]/(R - I (w - w0)))]}};                                                                                                   
 ODE /. YOURSsolution // FullSimplify

Thanks for pointing out mistakes. I am not rigorous. but it satisfied the differential equation if you calculate by hand. I have got the form I expected. I confused the general solution and special solution.

In[16]:=  ODE = {y'[t] == (I*w0 - R)*y[t] + B/2*r*(Exp[I*w*t] + Exp[-I*w*t])};
    sol = DSolve[ODE, y, t] // FullSimplify
Out[17]:={{y -> Function[{t}, 
    1/2 B E^(t (-R + I w0) + t (R - I (w + w0)))
       r (E^(2 I t w)/(R + I (w - w0)) + 1/(R - I (w + w0))) + 
     E^(t (-R + I w0)) C[1]]}}                                                                                                        
In[18]:= Simplify[1/2 B E^(t (-R + I w0) + t (R - I (w + w0)))
                 r (E^(2 I t w)/(R + I (w - w0)) + 1/(R - I (w + w0)))]                                                
 Out[19]:= 1/2 B E^(-I t w) r (E^(2 I t w)/(R + I (w - w0)) + 1/(R - I (w + w0)))

ODE = {y'[t] == (I*w0 - R)*y[t] + B/2*r*(Exp[I*w*t] + Exp[-I*w*t]), y[0] == 0};
sol = DSolve[ODE, y, t] // FullSimplify
ODE /. sol // FullSimplify(*Check if  Mathematica solution is correct. OK is correct. *)
(* {{True, True}} *)

YOURSsolution = {{y -> Function[{t}, (B*r)/2*(Exp[I w t]/(R + I*(w - w0)) + Exp[-I w t]/(
     R - I (w - w0)))]}};
ODE /. YOURSsolution // FullSimplify(*Check if  Yours solution is correct. It's seems is Not Correct !.Is not TRUE *)
 (* {{(B E^(-I t w) r w0)/(I R + w - w0) == 0, (B r R)/(R^2 + (w - w0)^2) == 0}} *)

I had corrected it ,thanks~