Subroutine euler2m(d1x,d2x,ti,xi,vi,tf,xf,vf)
c=======================================================
c euler2m.f: Solution of the second-order 1D ODE
c method:    modified Euler method (predictor-corrector)
c written by: Alex Godunov (October 2006)
c-------------------------------------------------------
c input ...
c d1x(t,x,v)- function dx/dt   (supplied by a user)
c d2x(t,x,v)- function d2x/dt2 (supplied by a user)
c ti 	- initial time
c xi  - initial position
c vi  - initial velocity (first order derivative)
c tf  - time for a solution
c
c output ...
c xf  - solution at point tf
c vf  - velocity at point tf
c=======================================================
      Real*8 d1x,d2x,ti,xi,vi,tf,xf,vf
      xf = xi + d1x(t,xi,vi)*(tf-ti)
      vf = vi + d2x(t,xi,vi)*(tf-ti)
c* correction
      xf = xi + (d1x(t,xi,vi)+d1x(t,xf,vf))*0.5*(tf-ti)
      vf = vi + (d2x(t,xi,vi)+d2x(t,xf,vf))*0.5*(tf-ti)
      Return
      End