The Riccati differential equation is a special form of a first order nonlinear differential equation and has the form:
y′(x) = f(x) ⋅ y^{2}(x) + g(x) ⋅ y(x) + h(x)
with the initial value
y(x_{0}) = y_{0}
where f(x), g(x) and h(x) are continuous functions on an interval I. The solution of the Riccati differential equation is generally difficult to find unless one knows a special solution. However, there are certain methods that can be used to find certain special solutions. The Riccati differential equation appears in many areas of mathematics and engineering, especially in control theory and differential geometry.
The solution of the Riccati differential equation is solved numerically. The used method can be selected. Three Runge-Kutta methods are available: Heun, Euler and RK4. The initial value can be varied by dragging the red point on the solution curve. In the input fields for f, g and h, up to three parameters a, b and c are used which can be varied by means of the slider in the graphics.
f(x)=
g(x)=
h(x)=
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 |
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ODE first order General first order ODE ODE second order ODE-System 2x2 ODE-System 3x3 Exponential growth Logistic growth Bernoulli equationCalculator for single ODEs:
y'+ay=b y'=fy^2+gy+h y'+2xy=xe^(-x^2) y'+xy=x y'+y=x y'=y^2 y'+y^2=1 y'=(Ay-a)(By-b)