c  test of euler method
      real x,y,dx,x0,y0
      real yreal,error
      pi=3.1415926
      x0=0.0                                                            
      y0=0.0
      xlim=1.0
      write(6,*) ' dx ?'
      read(5,*) dx
      n=xlim/dx
      nwrite=n/10
      i=0
      s2=0.0
      x=x0                                                              
      y=y0                        
   10 continue
      i=i+1
      call euler(x,y,dx)                                               
      yreal=sin(2.0*pi*x)
      error=y-yreal
      s2=s2+error**2
      write(2,601) x,y,yreal,error                                             
      if(mod(i,nwrite).eq.0) then
        write(6,601) x,y,yreal,error                                           
      endif
      if(x.gt.xlim) then
        s2=s2/i
        s=sqrt(s2)                                                 
        write(6,*) i,s
        stop
      endif
      go to 10                                                          
  601 format(1h ,4(1pe12.5,2x))                                         
      end                                                               
*********************************************************************** 
      function f(x,y)                                                   
      pi=3.1415926
      f=2.0*pi*cos(2.0*3.1415926*x)                                           
      return                                                            
      end                                                               
*********************************************************************** 
c  subroutine of euler method of one variable                    
      subroutine euler(x,y,dx)                                         
      real dx,dy,x,y
c                                                                       
      dy=f(x,y)*dx
c                                                                       
      x=x+dx                                                            
      y=y+dy                                                            
      return                                                            
      end