I suppose it is the integral part, because when I do this, I'm getting this as the solution.

    pdsolve(eq, hint= '1st_linear_constant_coeff_general_Integral')
    f(x, y) == (F(-b*y + c*x) + Integral(G((b*(b*x + c*y) + c*(-b*y + c*x))/(b**2 + c**2), (-b*(-b*y + c*x) + c*(b*x +   c*y))/(b**2 + c**2))*exp(a*(b*(b*(b*x + c*y) + c*(-b*y + c*x))/(b**2 + c**2) + c*(-b*(-b*y + c*x) + c*(b*x + c*y))/(b**2 + c**2))/(b**2 + c**2)), E)/(b**2 + c**2))*exp(-a*(b*x + c*y)/(b**2 + c**2))