A Cauchy – Euler equation is any equation of the form
where
and
is a continuous function. The solution is written as the sum of two terms:
The solution
of the homogeneous equation
(1) and a particular solution
of the non – homogeneous equation
where the form ofdepends onand may be found using guesswork and intuition. If we have two boundary conditions then we can solve fro any constant to find the general solution.
To find the solutionassume a solution of the form
Substitute these into (1).
Simplify and factorise with
to obtain
If we assume
then
This is called the indicial equation.
We can solve the above indicial equation in
to obtain
andhence 
Example: Fund the solution to
(3) if
and
when
The indicial equation is
We can solve this equation by factorising to obtain
Hence
or
for (3) so we do not need to look for a particular solution.
(4)
(5)
(4)+3*(5) gives
then from (4)
The solution is