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(in)dependent variables in DSolve: need explanation.


Hello,

I have a partial differential equation which I can solve by hand, but not yet in Mathematica. I need an explanation:
In queueing theory I have: z dP(z,t)/dt = (1-z)((a - b z)* P(z,t) - bP0(t)) with P(z,0)=z^i (i given). I can take the Laplace transform of the DE: 
(z^(i+1) - b (1-z)Laplace(P0(t))
---------------------------------
(sz - (1-z)(b - az)

The numerator must be zero, so it follows that Laplace(P(0,t)) is known now. The rest is substitution in the first formula and after that taking the inverse.

In fact I have the DE and search for solutions such that P0(t) exists. How do I do that in Mathematica with the DE and without applying the Laplace transform? Is that possible? 

The DE comes from the well known Markov equation for a single channel Poisson input, and exponential holding time. Its equation starts with dPo/dt = -a Po(t) + b P1(t) and the rest: dPn/dt=aPn-1(t)-(a+b)Pn(t)+bPn+1(t). 
The above used P(z,t) is the generating function of Pn:
P(z,t) = Sum from zero to infinity (Pn(t) z^n).
Well one can find it in Saaty's Elements of Queueing Theory, page 88. It is a standard procedure. I now would like to do it in Mathematica, and my basic question is: can it be done with DSolve only.


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