'PROGRAM WRITTEN BY INGEMAR BJERLE MARCH 2002
' SOLVING OF SECCOND ORDER LINEAR DIFFERENTIAL EQUATION
' INITIAL BOUNDARY VALUE SITUATION IS HANDLED

    [START]
       INPUT "NUMBER OF STEPS ";N
        PRINT"  X         Y"
           XIN=0
          XEND=5
           YIN=2
           UIN=0
         XEND1=0
      FOR K=0 TO 10
          DX=(XEND1-XIN)/N
          X=XIN:  Y=YIN:  U=UIN
          FOR I=1 TO N
           GOSUB [RUNGE]
          NEXT I
          PRINT X;"       "; Y
          XEND1=XEND1+XEND/10
       NEXT K

        CONFIRM" DO YOU WANT TO QUIT Y/N";answer$
        IF answer$="yes" THEN
        GOTO [END]
        ELSE
        GOTO [START]
        END IF
     [END] PRINT " ENTER TO END"
        INPUT R$ 
        PRINT " *** END ***"
        END

     [EQ]       '  SUBROUTINE EQUATIONS
        REM  DIFF EQ:  D2Y/DX2 + 2*DY/DX + 4*Y
        REM  WRITTEN AS TWO FIRST ORDER:  U=DY/DX:     DU/DX+2*U+4*Y
        REM  BOUNDARY CONDITIONS:  YIN=2:   UIN=1: XIN=0: XEND=5
       '***************************CHOOSE THE SAME VARIABLE NAMES AS IN RUNGE
        A=U
        B=(2*U+4*Y)*(-1)
       '***************************INSERT THE BOUNDARY CONDITIONS IN MAIN PROGR
        RETURN

     [RUNGE]      ' SUBROUTINE RUNGE
        rem  VARIABLES IN SUBROUTINE: INDEPENDENT: X  DEPENDENT:  Y AND U
        X=X: Y=Y: U=U

        GOSUB [EQ]
        K1=A*DX:    L1=B*DX

        X=X+DX/2: Y=Y+K1/2: U=U+L1/2
        GOSUB [EQ]
        K2=A*DX:    L2=B*DX
        X=X-DX/2: Y=Y-K1/2: U=U-L1/2

        X=X+DX/2: Y=Y+K2/2: U=U+L2/2
        GOSUB [EQ]
        K3=A*DX:    L3=B*DX
        X=X-DX/2: Y=Y-K2/2: U=U-L2/2

        X=X+DX:   Y=Y+K3:  U=U+L3
        GOSUB [EQ]
        K4=A*DX:    L4=B*DX
        X=X-DX:   Y=Y-K3:  U=U-L3

        X=X+DX
        Y=Y+K1/6+K2/3+K3/3+K4/6:   U=U+L1/6+L2/3+L3/3+L4/6
      RETURN