# Calculator for 2x2 differential equation systems 1.order

The differential equation system is given as follows:

ODE 1:  y1′ = f(x, y1, y2)

ODE 2:  y2′ = g(x, y1, y2)

## 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 and y02 can be varied with the sliders on the vertical axis at x0 in the first chart. The value for x0 can be set in the numeric input field. In the input fields for the functions f(x, y1, y2) and g(x, y1, y2), up to three parameters a, b and c can be used and changed by the sliders in the graph. The number of grid vectors in state-space diagram can be set in the numeric field for the grid points. In the state-space diagram is plotted y2 on the vertical axis and y1 about the horizontal axis.

↹#.000
🔍↔
🔍↕
Steps:
Method:
ODE 1: y1:
ODE 2: y2:

Axes ranges

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

Initial values

x0=
y01=
y02=

Parameter values

a=
b=
c=

Parameter ranges

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

Parameter ranges

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

### Solution in state space (phase space)

Grap the start point to move the initial values. Move the slider to see the grid at the given x value (time).

🔍↔
🔍↕
Grid points:
Scale grid=
Curve:
Grid vectors:

Axes ranges

y01-min=
y01-max=
y02-min=
y02-max=

y1′ = f(x, y1, y2) =

y2′ = g(x, y1, y2) =

cl
ok
Pos1
End
7
8
9
/
x
y1
y2
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 second order is:

y′′ = f(x, y, y′)

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

Substitution:

y1 = y

y2 = y′

So the resulting ODE system of 1.order is:

y1′ = y2

y2′ = f(x, y1, y2)

### Screenshot of the Image

Print or save the image via right mouse click.

## More Calculators

Here is a list of of further useful calculators:

Index Derivative calculus Partial derivatives and gradient Differential equations Exponential growth Logistic growth ODE first order General first order ODE ODE second order ODE-System 2x2 ODE-System 3x3 Gradient 2d Plot Function Plot