Markiyan Hirnyk, I'm not civilized, I'm scientist =D, but thanks a lot.

Thanks for all who tried to help me.

Perhaps, it's legit to do some kind of this: made a conclusion, that A(x,0)=0 implies Y(x,0)=0; then I0 expressed A(x,t) from formula with diff(Y(x,t),x$2) and substitute for rest one. As a result I recieved 1 pde:

diff(Y(x, t), x$2, t)*exp(2*x*b)+diff(Y(x, t), t)+diff(Y(x, t), x$2)=0. I forget to write '=0', but, as I know now, it is unnecessary.

Finally, I reached all my goals, by using plot, plot3d and numerical methods.

Common_file.mw