WOLFRAM|DEMONSTRATIONS PROJECT

Riccati Differential Equation with Continued Fractions

​
n
γ
α
k
xr
yr
Riccati's Equation
γ
2
y(x)
+α
′
y
(x)
k-2
x
3
2
y(x)
-3
′
y
(x)
3
x
-

3/2
x
J
-
4
5
2
5/2
x
5
3
3
J
1
5
2
5/2
x
5
3
-
1
x
+
4
x
-18+
3
5
x
-33+
3
5
x
-48+
3
5
x
-63+
3
5
x
-78+
3
5
x
-93+
3
5
x
-108-
5
x
36
Explore the solutions of the Riccati differential equation with continued fractions, which provide a very effective function approximation toolset. Usually the continued fraction expansion of a function approximates the function better than its Taylor or Fourier series. The solution(s) of the Riccati differential equation are very diverse; they contain polynomials, trigonometric and hyperbolic functions, logarithms, and (nested) square roots.