# 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.

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

Axes ranges

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

Initial values

x0=
y01=
y02=
y03=

Parameter values

a=
b=
c=

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
a⋅x+c / b⋅x+c
a+x / b+x
x2-a2/ x2+b2
a / x+b
1+√x / 1-√y
exsin(x)cos(x)
x+a
ea⋅x
a⋅x2+b⋅x+c
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
pow(a, b)Power ab
sqrt(x)Square root of x
exp(x)e-function
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 ...

## 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)

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