The differential equation system is given as follows:
ODE 1: y_{1}′ = f(x, y_{1}, y_{2}, y_{3})
ODE 2: y_{2}′ = g(x, y_{1}, y_{2}, y_{3})
ODE 3: y_{3}′ = h(x, y_{1}, y_{2}, y_{3})
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 y_{01}, y_{02} and y_{03} can be varied with the sliders on the vertical axis at x_{0} in the chart. The value for x_{0} can be set in the numeric input field. In the input fields for the functions f(x, y_{1}, y_{2}, y_{3}), g(x, y_{1}, y_{2}, y_{3}) and h(x, y_{1}, y_{2}, y_{3}), up to three parameters a, b and c can be used and changed by the sliders in the graph.
f(x,y_{1},y_{2},y_{3})=
g(x,y_{1},y_{2},y_{3})=
h(x,y_{1},y_{2},y_{3})=
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 |
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:
y_{1} = y
y_{2} = y′
y_{3} = y′′
So the resulting ODE system of 1.order is:
y_{1}′ = y_{2}
y_{2}′ = y_{3}
y_{3}′ = f(x, y_{1}, y_{2}, y_{3})
Print or save the image via right mouse click.
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