A treatise on the theory of Bessel functions, by G. N. Watson.

CHAPTER IV
DIFFERENTIAL EQUATIONS
4:1. Daniel Bernoulli's solution of Riccati's equation.
The solution given by Bernoulli* of the equation
~~~(1) ^ ^fdy
dz = az" + by2
consisted in shewing that when the index n has any of the values
0; _4 4 8 _. 12 _12 16 _ 16
I 35 3, 5? 5? - T7 7 9 — g *
while a and b have any constant valuest, then the equation is soluble by
means of algebraic, exponential and logarithmic functions. The values of n
just given are comprised in the formula
4m
(2) 4 m
2in + 1'
where m is zero or a positive integer.
Bernoulli's method of solution is as follows: If n be called the index of the
equation, it is first proved that the general equation+ of index n is transformable
into the general equation of index N, where
(3) N= n
n + 1'
and it is also proved that the general equation of index n is transformable
into the general equation of index v, where
(4) v = - 4.
The Riccati equation of index zero is obviously integrable, because the
variables are separable. Hence, by (4), the equation of index - 4 is integrable.
Hence, by (:3), the equation of index - 4 is integrable. If this process be continued by using the transformations (3) and (4) alternately, we arrive at the
set of soluble cases given above, and it is easy to see that these cases are
comprised in the general formula (2).
* Exercitationes quaedamn mathematicae (Venice, 1724), pp. 77-80; Acta Eruditorum, 1725,
pp. 473-475. The notation used by Bernoulli has been slightly modified; and in this analysis
n is not restricted to be an integer.
t It is assumed that neither a nor b is zero. If either were zero the variables would obviously
be separable.: That is, the equation in which a and b have arbitrary values.