Calculator for 3x3 differential equation systems 1.order

The differential equation system is given as follows:

ODE 1:  y1′ = f(x, y1, y2, y3)

ODE 2:  y2′ = g(x, y1, y2, y3)

ODE 3:  y3′ = h(x, y1, y2, y3)

Numerical solution of the ODE-System

The solution of the differential equations is calculated numerically. The used method can be selected. Three Runge-Kutta methods are available: Heun, Euler and Runge-Kutta 4.Order. The initial values y01, y02 and y03 can be varied with the sliders on the vertical axis at x0 in the chart. The value for x0 can be set in the numeric input field. In the input fields for the functions f(x, y1, y2, y3), g(x, y1, y2, y3) and h(x, y1, y2, y3), up to three parameters a, b and c can be used and changed by the sliders in the graph.

Scale:
Steps:
Method:
ODE 1: y1:
ODE 2: y2:
ODE 3: y3:

Initial values

x0=
y01=
y02=
y03=

Parameter values

a=
b=
c=

Axes ranges

x-min=
x-max=
y-min=
y-max=

Parameter ranges

a-min=
b-min=
c-min=

Parameter ranges

a-max=
b-max=
c-max=

f(x,y1,y2,y3)=

g(x,y1,y2,y3)=

h(x,y1,y2,y3)=

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ok
Pos1
End
7
8
9
/
x
y1
y2
y3
4
5
6
*
a
b
c
1
2
3
-
π
(
)
0
.
+
sin
cos
tan
ex
ln
xa
a/x
^
asin
acos
atan
x2
√x
ax
a/(x+b)
|x|
sinh
cosh

Transformation

The general ODE third order is:

y′′′ = f(x, y, y′, y′′)

With a substitution the differential equation of 3.order can be transformed to a differential system of first order.

Substitution:

y1 = y

y2 = y′

y3 = y′′

So the resulting ODE system of 1.order is:

y1′ = y2

y2′ = y3

y3′ = f(x, y1, y2, y3)

Usable expressions in the definition of the functions f, g and h

Constants

NameDescription
LN2Natural logarithm of 2
LN10Natural logarithm of 10
LOG2EBase 2 logarithm of EULER
LOG10EBase 10 logarithm of EULER
PIRatio of the circumference of a circle to its diameter
SQRT1_2Square root of 1/2
SQRT2Square root of 2

Trigonometric Functions

FunctionDescription
sin(x)sine of x
cos(x)Cosine of x
tan(x)Tangent of x
asin(x)arcsine
acos(x)arccosine of x
atan(x)arctangent of x
atan2(y, x)Returns the arctangent of the quotient of its arguments.
cosh(x)Hyperbolic cosine of x
sinh(x)Hyperbolic sine of x

Logarithm and Exponential

FunctionDescription
pow(b, e)e to the b
sqrt(x)Square root of x
exp(x)EULER to the x
log(x), ln(x)Natural logarithm
log(x, b)Logarithm to base b
log2(x), lb(x)Logarithm to base 2
log10(x), ld(x)Logarithm to base 10

More functions

FunctionDescription
ceil(x)Get smallest integer n with n > x.
abs(x)Absolute value of x
max(a, b, c, ...)Maximum value of all given values.
min(a, b, c, ...)Minimum value of all given values.
random(max = 1)Generate a random number between 0 and max.
round(v)Returns the value of a number rounded to the nearest integer.
floor(x)Returns the biggest integer n with n < x.
factorial(n)Calculates n!
trunc(v, p = 0)Truncate v after the p-th decimal.
V(s)Returns the value of the given element, e.g. sliders and angles.

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