The differential equation is given as follows:
ODE: y′′ = f(x, y, y′)
with the initial values
y(x_{0}) = y_{0} and y′(x_{0}) = y′_{0}
The solution of the general differential equation 2.order is calculated numerically. The method can be selected. Three Runge-Kutta methods are available: Heun, Euler and Runge-Kutta 4.Order. The initial values for y_{0} and y′_{0} can be varied by pulling the dots in the charts. The value for x_{0} can be set in numeric input field right. In the function up to three parameters a, b and c can be used which can be varied by means of the slider in the upper graph. In state-space diagram the solutions y_{1} and y_{2} of the corresponding general first order differential equation system are applied. The diagram shows y_{2} over y_{1}. 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 y_{2} on the vertical axis and y_{1} about the horizontal axis.
Grap the start point to move the initial values. The grid vectors show the initial direction if the ODE starts at this points.
f(x, y, ys) =
Function | Description |
---|---|
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 a^{b} |
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 |
With a substitution the differential equation of second order can be transformed to a differential system of first order.
Substitution:
y_{1} = y
y_{2} = y′
So the resulting ODE system of 1.order is:
y_{1}′ = y_{2}
y_{2}′ = f(x, y_{1}, y_{2})
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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