
(1) 
with initial conditions
, we assume the solution
takes the form
, i.e.,
, and convert the
differential equation above into an algebraic equation:

(2) 
where
is the characteristic polynomial. Solving
the equation we can find roots
in general,
and the solution of the differential equation can be found as a linear
combination:

(3) 
By letting
, we can find the coefficients
based on the initial conditions.
To solve an Nth order LCCDE

(4) 
with initial conditions
, we assume the solution
takes the form , i.e.,
, and convert the
difference equation above into an algebraic equation:

(5) 
where
is the characteristic polynomial. Solving
the equation we can find roots
in general,
and the solution of the difference equation can be found as a linear combination:

(6) 
By letting
, we can find the coefficients
based on the initial conditions.