# How to deal with first order complex differential equation?

I am trying to solve these two equations but find an error of fewer than the dependent variable and list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. I don't know what should I do? If anyone can resolve this, will be appreciated.

A1 = 1;
B1 = 0;
del0 = 1.50;
g0 = .0003;
dela = -1.5;(*Subscript[\[CapitalDelta], L] ; Laser-Lower polariton \
detuning*)
ome = 3.014;
ome1 = .125;
k1 = 0.1;
P0 = 20;
kex = .02;
kL = (k1 + kex)/2 - del0*(k1 - kex)/(2*ome);
Gma = 0.0001;
sol = DSolveValue[{a1'[t] - I*dela*a1[t] -
I*g0*(1 - del0/ome)*A1*(b1[t] + Conjugate[b1[t]])/2*a1[t] -
I*P0*(1 - del0/(2*ome))/Sqrt[2] -
I*P0*g0*B1*((b1[t] + Conjugate[b1[t]])/2)/(Sqrt[2]*ome) +
kL/2*a1[t] + (k1 - kex)*g0/ome*B1*(b1[t] + Conjugate[b1[t]])/2*
a1[t] == 0,
b1'[t] + I*ome1*b1[t] - I*g0*(1 - del0/ome)*A1*Abs[a1[t]]^2/2 -
I*P0*g0*B1*((a1[t] + Conjugate[a1[t]])/2)/(Sqrt[2]*ome) +
Gma*b1[t]/2 == 0, a1[t] == 1, b1[t] == 0}, a1, b1, t] // Chop
Plot[{Evaluate[Abs[a1[t]^2]] /. sol}, {t, 0, 500}]

• A couple of obvious problems. If you are trying to solve differential equations for a1[t] and b1[t] then giving it initial conditions a1[t]==1,b1[t]==0 sort of defeats the process. Probably you meant a1[0]==1,b1[0]==0 or something similar. Next, you finish up your DSolveValue with ...a1,b1,t] and that almost certainly needs to be ...{a1,b1},t] Next, for DSolve and DSolveValue giving it exact rational constants often enables it to select more powerful solution algorithms. But for a problem of this complexity you may want to use NDSolveValue instead, with the usual changes.
– Bill
Commented May 20, 2020 at 14:22
• @Thanks Bill. Fully satisfied with your answer.
– vini
Commented May 20, 2020 at 14:33
• Your use of Conjugate could be replaced by Re: FullSimplify[Re[z] == ((z + Conjugate[z])/2)]. As Bill hints, I'd be surprised if DSolve has algorithms to solve ODEs that have Conjugate or Re in them, so you probably have to use numerical methods. Commented May 20, 2020 at 15:01