Advances in Pure Mathematics, 2011, 1, 295-296
doi:10.4236/apm.2011.15053 Published Online September 2011 (http://www.SciRP.org/journal/apm)
Copyright © 2011 SciRes. APM
A Closed Form Solution to a Special Normal Form of
University College, The Pennsylvania State University, York, USA
Received May 18, 2011; revised June 28, 2011; accepted July 10, 2011
We present the general solution to the Riccati differential equation, 2
, for arbitrary real number
Keywords: Riccati Differential Equation, Bessel Functions
We consider a special normal form of Riccati equation,
z. We prove, for any real
number n, the equation has a closed analytic solution.
We show, the general solution for is a product
z and a combination of Bessel functions of vari-
ous n—dependent orders and arguments. We discuss the
variable transformations leading to this general result.
Introducing another set of variable transformations con-
ducive to a closed analytic solution, we by-pass the sin-
gular behavior of the Bessel functions for n = –2. We
show, the general solution in the limit of is
identical to the solution of n = –2.
Riccati equation is given by,
where , and are analytic functions of z.
It is known, for ,
transforms (1) into the normal form
It is the objective of this paper to solve (2) for a spe-
cial case where
. We prove the solution for any
real value of n is analytic.
We begin with Euler variable transformation, [1, p.
dd lnwzz yz . This linearizes (2)
Multiplying both sides of (3) by we compare the
14 0zyzv y
of [1, p. 206] and
deduce the following identities
Furthermore, according to the last reference, the solu-
tion of the given equation is,
is a solution of the Bessel equation of order v.
Therefore, we conclude the solution to, 22
where Jv and Yv are the Bessel functions of the first and the
second kind of order ν, with c1 and c2 being two arbitrary
constants. Substituting (4) in Euler transformation and
applying the chain differentiation, 1
with the prime notation indicating the derivative of the
Bessel functions with respect to variable