Look up the method of characteristics for first order PDE's. 

Thank You a lot,Kare! But I find a simple method of substitution of variables (x,t) by(y=x+Asin(t+phi,t). Then in the left side is simply dz/dy.But still not sure that everything  is true because the solution of the whole problem I solve  (nonstationary Schrodinger equation) is not physical.

What do You think about it?

You are somewhat off the track. One may say that the method of characteristics is a systematic way to find better coordinates, so the idea is basically right. But you must transform from two old variables to two new variables. Try

u = t, y = x -A*sin(t+phi) (note different sign in front of sin-term!)

This gives something like

dz/du = (i-u)*[y+A*sin(u+phi)]*B*cos(u+phi), and

z(u,y) = 1 + intergral((i-v)*[y+A*sin(v+phi)]*B*cos(v+phi), v=0..u)

I must say I am puzzled by the (t-i)-factor on the right hand side of your equation. But then I don't know the details of your problem, and how you approach it.

• The problem I try now to solve is quantum analogue of classical phenomenon of harmonic forced oscillations near the point of unstable equilibrium. This is theoretically possible (like the ball on the hill) when proper initial conditions on x(0) and p(0) being confirmed with initial phase phi exclude exponentially growing term. The set of such ICs is a line or interval in phase space. Really these ICs are always disturbed by some delta(x or p) and it results in finity of time tau before the oscillating body leaves finally the balance point area. In quantum mechanics if You define properly an initial wave function being essentially nonzero only on the set above You may receive some metastable state with the life time tau>>T - the period of oscillating homogeneous field exposed on such unstable system. That is the w.f. will stay compact during a long time.
• It may be an elrctron on destructive orbital level of  molecule rotating in strong electric field or proton being on highly excited level inside rotating nucleus.
• I suppose the solution of this problem may be deduced from analogous problem of quantum harmonic oscillator in external field oscillating with the same frequency.

Here is an alternate form which you could try to resolve into real and complex components

1/2 A e^(-i t-i phi) z'(x)+1/2 A e^(i t+i phi) z'(x)+z'(t) = -1/2 B (t-i) x (e^(-i t-i phi)+e^(i t+i phi))

see

Wolfram Alpha

which interpretes your equation as

z'(t)+z'(x) A cos(t+phi) = -(B (cos(t+phi) x (t-i))) and throws out not the solution but the above, which you could try to resolve into components (Re and Im).

Or you might find the above alternative not useful, but worth looking at Wolfram Alpha which throws out other alternatives.

Or try a steady state solution first by setting z'(t)=0. I read somewhere that there is a trick that can be used to get a solution to the time dependent Shrodinger equation, by first solving the steady state Shrodinger using Fourier Transforms. Might that be possible here or does i complicate thinks.

Just some ideas......

http://www.wolframalpha.com/input/?i=dz%2Fdt%2B%28dz%2Fdx%29*A*cos%28t%2Bphi%29%3D-B*cos%28t%2Bphi%29*x*%28t-i%29

2 > an initial wave function being essentially nonzero only on the set above

The least you need to do quantum mechanically is to satisfy the initial conditions

(1) < Q >0 = 0,  (2) < Q2 >0 = 0,

where Q is a hermitian operator. Equation (1) is the classical condition. Equation (2) only has a non-normalizable wavefunction, like

exp( iq2/2 ),  as solution. 

The last term of the equation seems to involve a shift along the imaginary time axis. So, first of all, this needs to be clarified, since this point doesn't seem to have been addressed and, next, this fact, alone, implies that the initial condition given isn't sufficient. It would be useful to define the domain and the range: it seems that z(x,t) is a complex-valued function of three variables, x, that, implicitly, seems to take real values, and t that, apparently, takes complex values. The shift operator, of course, contains derivatives of all orders, so this isn't, in fact a partial differential equation, but an operator equation.