**Casio fx-9750GIII and fx-CG50: System of Two Differential Equations, Runge Kutta 4th Order**

**Introduction**

The program TWORK4 uses the Runge Kutta 4th Order method to solve the following system of differential equations:

dx/dt = f(t, x, y)

dy/dt = g(t, x, y)

with initial conditions x0 = x(t0) and y0 = y(t0)

The next step is calculated with step h from:

x1 ≈ x0 + (k1 + 2 * k2 + 2 * k3 + k4) / 6

y1 ≈ y0 + (l1 + 2 * l2 + 2 * l3 + l4) / 6

k1 = h * f(t0, x0, y0)

l1 = h * g(t0, x0, y0)

k2 = h * f(t0 + h / 2, x0 + k1 / 2, y0 + l1 / 2)

l2 = h * g(t0 + h / 2, x0 + k1 / 2, y0 + l1 / 2)

k3 = h * f(t0 + h / 2, x0 + k2 / 2, y0 + l2 / 2)

l3 = h * g(t0 + h / 2, x0 + k2 / 2, y0 + l2 / 2)

k4 = h * f(t0 + h, x0 + k3, y0 + l3)

l4 = h * g(t0 + h, x0 + k3, y0 + l3)

For the next step set t0 = t0 + h, x0 = x1, and y0 = y1.

Inputs:

DX/DT: Enter dx/dt as a function of T, X, and Y. T is the independent variable.

DY/DT: Enter dy/dt as a function of T, X, and Y. T is the independent variable.

T0, X0, Y0: Enter the initial conditions

STEP: Enter the step size

ITERATIONS: Enter the number of iterations desired. This allows you to calculate a far point in one leap. Example, if your initial condition is to = 0 and you want to find the point when t = 1 using the step size of 0.1, enter 0.1 for STEP and 10 for ITERATIONS.

**Casio fx-9750GIII and fx-CG 50 Program: TWORK4**

Notes:

* Although the code for the two calculators are the same, the programs will need to be programmed on each calculator separately.

* The slash character (/) is accessed from the CHAR submenu. The submenu shows up at the top-level menu which would read:

TOP/BOTTOM/SEARCH/MENU/A ←→ a/CHAR

* fn1 and fn2 are stored as function memories.

"2020-06-18 EWS"

Rad

"DX/DT="? → fn1

"DY/DT="? → fn2

"T0"? → U

"X0"? → A

"Y0"? → B

"STEP?" → H

4 → Dim List 25

4 → Dim List 26

Lbl 0

"ITERATIONS"? → N

For 1 → I To N

A → X: B → Y: U → T

H * fn1 → List 25[1]

H * fn2 → List 26[1]

A + List 25[1] ÷ 2 → X

B + List 26[1] ÷ 2 → Y

U + H ÷ 2 → T

H * fn1 → List 25[2]

H * fn2 → List 26[2]

A + List 25[2] ÷ 2 → X

B + List 26[2] ÷ 2 → Y

H * fn1 → List 25[3]

H * fn2 → List 26[3]

A + List 25[3] → X

B + List 26[3] → Y

U + H → T

H * fn1 → List 25[4]

H * fn2 → List 26[4]

A + (List 25[2] + List 25[3] + Sum List 25) ÷ 6 → A

B + (List 26[2] + List 26[3] + Sum List 26) ÷ 6 → B

U + H → U

Next

ClrText

"(T, X, Y)"

{U, A, B} ◢

Menu "NEXT?", "YES", 0, "NO", 1

Lbl 1

"DONE"

**Example**

dx/dt = x sin y

dy/dt = y^2/500 + y/3 - y

Initial conditions: x(0) = 0.1, y(0) = 0.1, h = 0.1

T0 = 0

STEP = 0.1

ITERATIONS: 10

Results: (T, X, Y): (1, 0.1075645239, 0.05134921355)

ITERATIONS: 10

Results: (T, X, Y): (2, 0.1116714419, 0.02636554462)

Source:

John W. Harris and Horst Stocker.

__Handbook of Mathematics and Computational Science__. Springer: New York. 2006. ISBN 978-0-387-94746-4

Stack Exchange. "Help with using the Runge-Kutta 4th order method on a system of 2 first order ODE'S" Asked March 2014. Last updated February 2019. https://math.stackexchange.com/questions/721076/help-with-using-the-runge-kutta-4th-order-method-on-a-system-of-2-first-order-od Retrieved June 17, 2020

Eddie

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