A procedure for determining the behavior of an 
th order ordinary
differential equation at a removable singularity
without actually solving the equation. Consider
 |
(1)
|
where 
is analytic in 
and rational in its other arguments. Proceed by making the
substitution
 |
(2)
|
with 
.
For example, in the equation
 |
(3)
|
making the substitution gives
 |
(4)
|
The most singular terms (those with the most negative exponents) are called the "dominant balance terms," and must balance exponents and coefficients at the singularity. Here, the first two terms are dominant, so
 |
(5)
|
 |
(6)
|
and the solution behaves as 
. The behavior in the neighborhood
of the singularity is given by expansion in a Laurent series, in this case,
 |
(7)
|
Plugging this series in yields
 |
(8)
|
This gives recurrence relations, in this case with 
arbitrary, so the 
term is called the resonance or Kovalevskaya
exponent. At the resonances, the coefficient
will always be arbitrary. If no resonance term is present, the pole
present is not ordinary, and the solution must be investigated using a psi
function.
