' 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
        NN=10
      FOR K=0 TO NN
        DX=(XEND1-XIN)/N
        X=XIN:  Y=YIN:  U=UIN
          FOR I=1 TO N
            GOSUB [RUNGE]
          NEXT I
        PRINT USING("###.###",X);"       "; Y
        XEND1=XEND1+XEND/NN
      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=0: XIN=0: XEND=5
        'A=DY B=DU D=DZ
       '***************************CHOOSE THE SAME VARIABLE NAMES AS IN RUNGE
        A=U
        B=(2*U+4*Y)*(-1)
        'D=
       '***************************INSERT THE BOUNDARY CONDITIONS IN MAIN PROGR
        RETURN

     [RUNGE]      ' SUBROUTINE RUNGE
         rem  VARIABLES IN SUBROUTINE: INDEPENDENT: X  DEPENDENT:  Y, U AND Z
        X=X: Y=Y: U=U: Z=Z
        GOSUB [EQ]
        K1=A*DX:    L1=B*DX:    F1=D*DX

        X=X+DX/2: Y=Y+K1/2: U=U+L1/2: Z=Z+F1/2
        GOSUB [EQ]
        K2=A*DX:    L2=B*DX:    F2=D*DX
        X=X-DX/2: Y=Y-K1/2: U=U-L1/2 :Z=Z-F1/2

        X=X+DX/2: Y=Y+K2/2: U=U+L2/2 :Z=Z+F2/2
        GOSUB [EQ]
        K3=A*DX:    L3=B*DX:    F3=D*DX
        X=X-DX/2: Y=Y-K2/2: U=U-L2/2 :Z=Z-F2/2

        X=X+DX:   Y=Y+K3:  U=U+L3    :Z=Z+F3
        GOSUB [EQ]
        K4=A*DX:    L4=B*DX:    F4=D*DX
        X=X-DX:   Y=Y-K3:  U=U-L3    :Z=Z-F3

        X=X+DX
        Y=Y+K1/6+K2/3+K3/3+K4/6:   U=U+L1/6+L2/3+L3/3+L4/6
        Z=Z+F1/6+F2/3+F3/3+F4/6
      RETURN