The differential equation system is given as follows:
ODE 1: y1′ = f(x, y1, y2)
ODE 2: y2′ = g(x, y1, y2)
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.
Grap the start point to move the initial values. Move the slider to see the grid at the given x value (time).
y1′ = f(x, y1, y2) =
y2′ = g(x, y1, y2) =
|sin(x)||Sine of x|
|cos(x)||Cosine of x|
|tan(x)||Tangent of x|
|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|
|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|
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.
y1 = y
y2 = y′
So the resulting ODE system of 1.order is:
y1′ = y2
y2′ = f(x, y1, y2)
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