PROGRAM example20
!  Runge-Kutta method
! will only do first half of question
   IMPLICIT NONE
   REAL :: x,y,h,k1,k2,k3,k4,error
   REAL :: x1,y1,xmax
   INTEGER :: I,J,jval,jprint
   INTERFACE
   FUNCTION func(x,y)   
   IMPLICIT NONE
   REAL, INTENT(IN) :: x,y
   REAL :: func
   END FUNCTION func

  FUNCTION sol(x)
  IMPLICIT NONE
  REAL :: sol
  REAL , INTENT(IN):: x
   END FUNCTION sol
 END INTERFACE
   PRINT *, 'enter step size and  maximum x value'
   READ *, h,xmax
   x=0
   y=1
   error=y-sol(x)
   WRITE(6,1000)x,y,error
1000 FORMAT(5(E12.5,1X))
   jval=1
!  the jval count when we get to integer values of x
! jprint is how many h steps to reach x=1
   jprint=NINT(1./h) 
   DO WHILE (x <= xmax)

   k1=func(x,y)
   x1=x+h*.5
   y1=y+h*k1*.5
   k2=func(x1,y1)
   y1=y+h*k2*0.5
   k3=func(x1,y1)
   x1=x+h
   y1=y+h*k3
   k4=func(x1,y1)
   y=y+(h/6.)*(k1+2.*k2+2.*k3+k4)
   x=x1
   error= y-sol(x)
! check if x is close to integer value for printing
    if(jval==jprint)THEN

    WRITE(6,1000)x,y,error
! now reset counter jval
    jval=0
   ENDIF

   jval=jval+1
  ENDDO
  END PROGRAM example20
  FUNCTION func(x,y)   
  IMPLICIT NONE
  REAL, INTENT(IN) :: x,y
  REAL :: func
  func = y*0.25*(1.-(y/20.))
  END FUNCTION func

  FUNCTION sol(x)
  IMPLICIT NONE
  REAL :: sol
  REAL , INTENT(IN):: x
  sol=20./(1.+ 19.*EXP(-x*.25))
  END FUNCTION sol