FullSimplify[ DSolve[{y'''[z] == -I*Betta*y'[z] + B}, y[z], z] /.
Solve[0 ==
 C[3] + C[2]*Cosh[I/2*(I*Betta)^0.5] +
  C[1]/I*Sinh[I/2*(I*Betta)^0.5] + B/(2*I*Betta) &&
0 == C[3] + C[2]*Cosh[I/2*(I*Betta)^0.5] -
  C[1]/I*Sinh[I/2*(I*Betta)^0.5] - B/(2*I*Betta) &&
1 == C[2]*I*(I*Betta)^0.5*Sinh[I/2*(I*Betta)^0.5] +
  C[1]*(I*Betta)^0.5*Cosh[I/2*(I*Betta)^0.5] + B/(I*Betta) &&
1 == -C[2]*I*(I*Betta)^0.5*Sinh[I/2*(I*Betta)^0.5] +
  C[1]*(I*Betta)^0.5*Cosh[I/2*(I*Betta)^0.5] + B/(I*Betta), {C[1],
 C[2], C[3], B}] ]// TraditionalForm