program rk4sysmain
!___________________________________________________________

! This program solves the initial value problem for a system
! of first-order differential equations using the classical
! fourth-order Runge-Kutta method.
!___________________________________________________________

! Declare problem size:
integer, parameter :: m=?

! Declare program variables:
real :: mass,...
integer :: n, ...
real :: t0, ...
real, dimension(m) :: y, ...
character(len=20) :: datafile, ??????
namelist /??????????/ mass,...
???????? /run_data/ t0,...,y

!...Open and enter parameters from file parameters.dat:
open (unit=7, file='parameters.dat', status='old')
read (7,nml=parameters)
close(7)

!...Echo parameters to terminal:
write (*,nml=parameters)

!...Name, open and enter run_data from data_file:
write (*,*) ' Enter run_data file name:'
read (*,10) ????????
10 format (a)
open (????=7, file=datafile, status='old')
read (7,nml=...)
close(7)

!...Echo run_data to terminal:
write (*,???=run_data)

! Name, open and head output file:
write (*,*) ' Enter output file name:'
read (*,10) output
open (unit=8, file=??????, status='unknown')
write (8,30)
30 format('   t       y           y''  ')  !Q1 (adjust spacing if needed)
!30 format('  t       x     ...            !Q2

! Initialise values for DO-loop (y initialised directly by input):
t = ...
h = ...

! Initialise min_r1 for Q2 only:
! r1_min = ...  !Q2

t_loop: do j= 1, ?
   ! 4th order Runge-Kutta subroutine:
   call rk4
   y = ????
   t = ...
   ! Print solutions every nprint steps:
   if ( mod(j,nprint)==0 ) then
!      r1 = ...  !Q2
!      write (8,40) t, ..., r1  !Q2
      write (8,40) t, ...
40    format (?????, ??????)
   endif

   ! The following code is for Question 2 only:
   !r1 = ...
   !Calculate the minimum of r1:
   !if (r1_min > r1) then
   !   r1_min= ...
   !   .
   !   .
   !   .
   !endif

enddo t_loop

! The following ... is for Question 2 only:
! Output the minimum of r1 and associated values of t,y:
   !   .
   !   .
   !   .

stop

????????
!-----------------------------------------------
function f(t,y) result(f_result)
real, intent(in):: t
real, dimension(m), intent(in):: y
real, dimension(m) :: f_result

f_result(1) = ...
   !   .
   !   .
   !   .

end ???????? ?
!-----------------------------------------------------------------
subroutine rk4
   ! Purpose:  To implement the classical fourth-order Runge-Kutta
   ! method for the system of M first-order differential equations
   ! y'=f(t,y) over one interval of width h.  Thus subroutine rk4
   ! approximates y(t+h) with yrk4, given y=y(t).

k1 = h*f(t,y)
k2 = h*f(t + h/2.,y + k1/2.)
k3 = h*f(t + h/2.,y + k2/2.)
k4 = h*f(t+h,y+k3)
yrk4 = y + (k1 + 2.*k2 + 2.*k3 + k4)/6.

return
end subroutine rk4
!-----------------------------------------------------------------
end ...