Sharp Ordinary Differential Equations, Calculation, EL-9900 Graphing Calculator, Flowchart

EL-9900 Graphing Calculator

Ordinary Differential Equations

Enter the initial conditions (X, Y) with the step H and interval T. Use Runge Kutta Gill method to solve the ordinary differential equation of first order.

Calculation

Use the following four steps of Runge Kutta Gill method to find the equation Xn + 1 and Yn - 1 from Xn and Yn. Input Qo = 0 at the

starting point X0. Y

1.K0 = Hf (Xn , Yn), R1 = (1/2) (K0-2Q0), Y (1)= Yn +R1

2.Q1 = Q0 + 3R1- (1/2)K0

K1 = Hf (Xn + H/2, Y(1)), R2 = (1 - 1/2) (K1-Q1), Y(2)=Y(1)+ R2			Y3
3. Q2 = Q1 + 3R2 - (1 -		K1	Y2
K2	= Hf (Xn + H/2, Y(2)), R3 = (1 + 1/2) (K2 -Q2),Y(3)= Y(2)+ R3		Y1
4. Q3	= Q2 + 3R3- (1 + 1/2) K2			h	h
						X
K3	= Hf (Xn+1, Y(3)), R4 = (1/6) (K3-2Q3), Yn+1 = Y(3)+ R4		0	X1	X2 X3
Q4	= Q3 + 3R4 - (1/2)K3

FLOWCHART

Start

Entry of data

Initial setting

MAIN

Gosub

Calculation of step 1.

Gosub

Enter Data.

Initial coordinates (X, Y), step

of x (H), and interval of solutions (T) Data for calculation set.

Calculation executed.

Jumps to subroutine.

Jumps to subroutine.

Subroutine

FORMULA Subroutine for

calculating built-in function

Return

Subroutine for calculating built-in function

f = -I J

(Another equation can be used.)

Calculation of step 2.

Gosub

Calculation of step 3.

Gosub

Jumps to subroutine.

Jumps to subroutine.

PROGRAMME LIST(REAL MODE)

Title : RUNGE

Rem INITIAL	I+H/2 I	Goto MAIN
Print " Input X0	Rem 2	Label NEXT
Input X	Gosub FORMULA	If Z≠ I Goto SUB2
Print " Input Y0	H F K	I M

Calculation of step 4.

	N	Judgment of calculation end.
Z <= I		If calculation result of I smaller
Y		than value of increase of I,
		calculation repeated again.
S = I
O = J
Z ≠ I	N		Following calculation
	SUB2		performed when calculation
Y
	Processing		result of x not equal to the
			value of increase of X.
	in case of			(Z - S) (J - O)		M = Z
	inequality		P =		+ O,
				H		N = P

SUB1

Display of result

Processing for	Prior processing for next calculation
next calculation	Z = Z + T, S = X, O = J

Input Y				B (K-Q) R				J	N
X	I			J+R		J		Label SUB1
Y	J			Q+3		R-B K Q		ClrT
Print " Input H				Rem 3				Print "XN=
Input H				Gosub FORMULA				Print M
Print " Input T				H	F	K		Print "YN=
Input T				A	(K-Q) R			Print N
1+√	(2-1) A			J+R		J		Wait
1- √	(2-1) B			Q+3		R - A K Q		Z+T Z
I+T		Z		I+H/2 I				I	S
O	Q			Rem 4				J	O
I S				Gosub FORMULA				Goto MAIN
Label MAIN				H F		K		Label SUB2
Rem 1				(K - 2 Q) /6			R	(Z-S) (J-O) /H+O P
Gosub FORMULA				J+R		J		Z	M
H F		K		Q+3		R - K/2	Q	P	N
(K-2		Q) /2	R	If Z≤ I Goto NEXT				Goto SUB1
J+R		J		I	S			Label FORMULA
Q+3 R-K/2			Q	J	O			-I J F
								Return

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