# Calculator for 2x2 differential equation systems 1.order

ODE 1:  y1′ = f(x,y1,y2)

ODE 2:  y2′ = g(x,y1,y2)

f(x,y1,y2) =
 cl ok Pos1 End 7 8 9 / x y1 y2 4 5 6 * π ( ) 1 2 3 - a b c 0 . + sin cos tan ex ln log10 asin acos atan x2 √x xy |x| sinh cosh
g(x,y1,y2) =
 cl ok Pos1 End 7 8 9 / x y1 y2 4 5 6 * π ( ) 1 2 3 - a b c 0 . + sin cos tan ex ln log10 asin acos atan x2 √x xy |x| sinh cosh

Direction field:
Numerics:

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

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

x0=
y01= y02=

a=
b=
c=

## Numerical solutuion of the ODE-System

The solution of the differential equations is calculated numerically. The method can be selected. Three Runge-Kutta methods are available: Heun, Euler and Runge-Kutta 4.Order. The initial values Y01 and Y02 can be varied with the sliders in the first chart. The value for x0 can be adjusted in Numericsfeld. In the input fields for f and g, up to three parameters a, b and c are used which can be varied by means of the slider in the upper graph. The number of calculated solutions in phase space diagram can be set in Numericsfeld under Step size Direction field. In the phase-space diagram is plotted on the vertical axis y2 y1 about the horizontal axis.

### Usable expressions in the definition of the functions f and g

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.