[A] and [B] are matrices? what size they have?

are they known? what do you mean by exp([B])?

Try making the equation a first order differential so as to get a particular solution and the analytic one. Unfortunately I wont go into details on how to do it

I would like to see the equation.  If it is nonlinear because of the operator coefficients, find a functional relationship for them that will make the system linear for say a time step. 

Try quasilinear methods as published by V. Mandelzweig et al., which can be adapted to nonlinear matrix equations.

The basic idea therein is based on Newton-Cotes method.

Faisal's answer does not take the steps needed for rapid convergence to a solution. The Newton-Raphson  method and Mandelzweig's  QLM method do iterate,  but also generate a linear intermediate equation which serves to guide the answer to rapid convergence.

Without that last step,  convergence will be either slow, nil,  or just random.

Note:  I must correct myself: it is the Newton-Raphson method  not Newton-Cotes

Dear Liu,

   I suggest you to go for any iterative procedure that needs some initial conditions which must be have convergence status and stability too. Then only we can finalize the appropriate scheme. Otherwise, we can make use of MATLAB Solver.

Dear Dr. Liu:

For solving those differential equations in matrix form I suggets using Matlab or Mathematica.

Cheers Humberto