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

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.