where
and
is
a continuous function. The solution is written as the sum of two
terms:
University Maths Notes: Calculus – Cauchy - Euler Equations (1)
A Cauchy – Euler equation is any equation of the
formwhere
andis
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
