# Calculator Riccati differential equation

## Riccati differential equation

The Riccati differential equation is a special form of a first order nonlinear differential equation and has the form:

y′(x) = f(x) ⋅ y2(x) + g(x) ⋅ y(x) + h(x)

with the initial value

y(x0) = y0

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.

### Calculator for the initial value problem of the Riccati equation with the initial values x0, y0

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.

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Method:
Steps:
Grid points:
Scale grid:
Curve:
Grid:
f(x):
g(x):
h(x):

Axes ranges

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

Initial values

x0=
y0=

Parameter value

a=
b=
c=

Parameter ranges

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

f(x)=

g(x)=

h(x)=

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End
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a
b
c
1
2
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-
π
(
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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 ...

### Screenshot of the logistc growth graph

Print or save the image via right mouse click.

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