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What I wrote to combine them is this:

    Program RK4
    implicit none

    real k1,k2,k3,k4,h,t,R
    integer i,n
    real a
    read*,n,h 


    t=0
    R=0

    Do i=1,n

      k1=h*(1/a(t))

      k2=h*(1/a(t+h/2.0))

      k3=h*(1/a(t+h/2.0))

      k4=h*(1/a(t+h))

      t=t+h

      R=R+(k1+2*k2+2*k3+k4)*(1/6.0)

      write(*,*)t,R

    End Do

    end program

    !-----------------------------------------

    SUBROUTINE Scale_Factor(h,n,t,a)
      implicit none
      real t,a,k1,k2,k3,k4,h,g
      integer i,n

      t=0
      a=0.001


    Do i=1,n

       k1=h*g(a)

       k2=h*g(a+k1/2.0)

       k3=h*g(a+k2/2.0)

       k4=h*g(a+k3)

       t=t+h

       a=a+(k1+2*k2+2*k3+k4)*(1/6.0)

       write(*,*)t,a

    END DO
    END SUBROUTINE

    !-------------------------
    FUNCTION g(a)
      implicit none
      real a,g
      g=sqrt((1.0/a)+(1.0/a**2)) 
    END FUNCTION

But I know it is not correct.

I want to solve this integral using RK4 by coding in Fortran:

$$R=∫1/a(t) dt → dR/dt=1/a(t) =f(t)$$

Initial point: t=0 (or a=0.001) and R=0

And I have to get a(t) by solving another differential equation: $$da/dt=1/a+1/a^2 =g(a)$$

Initial point: t=0 and a=0.001

I wrote this code to get a(t):

    PROGRAM RK4
      implicit none
      real h,t
      integer n
      read*,h,n
      call Scale_Factor(h,n,t,a)
    END PROGRAM

    !---------------------------------------------

    SUBROUTINE Scale_Factor(h,n,t,a)
      implicit none
      real t,a,k1,k2,k3,k4,h,g
      integer i,n

      t=0
      a=0.001


    Do i=1,n

       k1=h*g(a)

       k2=h*g(a+k1/2.0)

       k3=h*g(a+k2/2.0)

       k4=h*g(a+k3)

       t=t+h

       a=a+(k1+2*k2+2*k3+k4)*(1/6.0)

       write(*,*)t,a

    END DO
    END SUBROUTINE

    !-------------------------
    FUNCTION g(a)
      implicit none
      real a,g
      g=sqrt((1.0/a)+(1.0/a**2)) 
    END FUNCTION

And I have another similar program for solving the first integral. But I need to use a(t) that this program produces to solve the integral. I do not know how to do it.

I want to solve this integral using RK4 by coding in Fortran:

$$R=∫1/a(t) dt → dR/dt=1/a(t) =f(t)$$

Initial point: t=0 (or a=0.001) and R=0

And I have to get a(t) by solving another differential equation: $$da/dt=1/a+1/a^2 =g(a)$$

Initial point: t=0 and a=0.001

I wrote this code to get a(t):

    PROGRAM RK4
      implicit none
      real h,t
      integer n
      read*,h,n
      call Scale_Factor(h,n,t,a)
    END PROGRAM

    !---------------------------------------------

    SUBROUTINE Scale_Factor(h,n,t,a)
      implicit none
      real t,a,k1,k2,k3,k4,h,g
      integer i,n

      t=0
      a=0.001


    Do i=1,n

       k1=h*g(a)

       k2=h*g(a+k1/2.0)

       k3=h*g(a+k2/2.0)

       k4=h*g(a+k3)

       t=t+h

       a=a+(k1+2*k2+2*k3+k4)*(1/6.0)

       write(*,*)t,a

    END DO
    END SUBROUTINE

    !-------------------------
    FUNCTION g(a)
      implicit none
      real a,g
      g=sqrt((1.0/a)+(1.0/a**2)) 
    END FUNCTION

And I have another similar program for solving the first integral. But I need to use a(t) that this program produces to solve the integral and I do not know how to combine them in a single program.

I want to solve this integral using RK4 by coding in Fortran:

$$R=∫1/a(t) dt → dR/dt=1/a(t) =f(t)$$

Initial point: t=0 (or a=0.001) and R=0

$$k_1=hf(t_0,R_0 )=h 1/(a(t_0))$$ $$k_2=hf(t_0+h/2,R_0+k_1/2)=h 1/(a(t_0+h/2))$$ $$k_3=hf(t_0+h/2,R_0+k_2/2)=h 1/(a(t_0+h/2))$$ $$k_4=hf(t_0+h,R_0+k_3 )=h 1/a(t_0+h)$$ $$R_1=R_0+ 1/6(k_1+2k_2+2k_3+k_4)$$

And I have to get a(t) by solving another differential equation: $$da/dt=1/a+1/a^2 =g(a)$$

Initial point: t=0 and a=0.001

$$k_1=hg(t_0,a_0 )$$ $$k_2=hg(t_0+h/2,a_0+k_1/2)$$ $$k_3=hg(t_0+h/2,a_0+k_2/2)$$ $$k_4=hg(t_0+h,a_0+k_3 )$$ $$a_1=a_0+ 1/6(k_1+2k_2+2k_3+k_4)$$

I can write the code of RK4 method in Fortran but I have problem in combining these two Rk4 programs. I need to write a code that solves the second differential equation and produces a(t) and then uses it in solving the first integral. Can anyone help me?

I want to solve this integral using RK4 by coding in Fortran:

$$R=∫1/a(t) dt → dR/dt=1/a(t) =f(t)$$

Initial point: t=0 (or a=0.001) and R=0

And I have to get a(t) by solving another differential equation: $$da/dt=1/a+1/a^2 =g(a)$$

Initial point: t=0 and a=0.001

I wrote this code to get a(t):

    PROGRAM RK4
      implicit none
      real h,t
      integer n
      read*,h,n
      call Scale_Factor(h,n,t,a)
    END PROGRAM

    !---------------------------------------------

    SUBROUTINE Scale_Factor(h,n,t,a)
      implicit none
      real t,a,k1,k2,k3,k4,h,g
      integer i,n

      t=0
      a=0.001


    Do i=1,n

       k1=h*g(a)

       k2=h*g(a+k1/2.0)

       k3=h*g(a+k2/2.0)

       k4=h*g(a+k3)

       t=t+h

       a=a+(k1+2*k2+2*k3+k4)*(1/6.0)

       write(*,*)t,a

    END DO
    END SUBROUTINE

    !-------------------------
    FUNCTION g(a)
      implicit none
      real a,g
      g=sqrt((1.0/a)+(1.0/a**2)) 
    END FUNCTION

And I have another similar program for solving the first integral. But I need to use a(t) that this program produces to solve the integral. I do not know how to do it.