This program computes the solution of second order linear
ordinary differential equations by the method of Frobenius (substitution
of infinite series.) The program is called by:
LOADFILE(SERIES,FASL,DSK,SHARE);
SERIES(eqn,y,x);
where "eqn" is a differential equation of the form:
f(x)*diff(y,x,2)+f'(x)*diff(y,x)+f''(x)*y=0
and "y" and "x" are the dependent and independent variables.
The dependencies must be established ahead of time by:
DEPENDS(Y,X);
The program will determine the recurrence relation and if that
is a function of only one subscripted variable, will go on to determine
a complete solution in terms of an infinite series or a sum of them. If
the recurrence relation is more complex;e.g.,
A[N]=A[N-1]+A[N-2]
then the relation will be printed and the program will produce a
truncated series solution using the global variable NUMTERMS[5].
These truncated solutions courtesy of JF...OLVER >.
The program produces answers containing some compact functions,
namely,FFF( ) and HARM( ). These are defined in Rainville:Intermediate
Differential Equations (pp 109-112) as follows:
FFF(exp,n)=exp(exp+1)(exp+2)...(exp+n-1) n ge 1
and FFF(exp,0)=1, exp#0
and HARM(exp,n) is the partial sum of the harmonic series:
n
====
\ 1
HARM(exp,n) = > -------
/ exp + k
====
k = 1
and the special case:
HARM(1,n) = SUM(1/k,k,1,n)
Several variables and flags are available to the user:
VERBOSE1[TRUE] if FALSE will print diagnostic information about the equation,
i.e. the point at which the solution is being found (0), is an ord-
inary point, singular point or the roots of the indicial equation
are equal or differ by an integer and therefore there will be a
logarithmic term in the solution.
POINTEXPAND[0] if set to any value other than zero will cause the diagnostics
to try that point but will not solve the equation there (yet)!!
NUMTERMS[5] if set to another value will cause the diagnostic routine to
compute a different number of terms for the Taylor coefficients
when determining the nature of the singularities and will
truncate an approximate solution(if necessary) at that value.
The program will not work for expressions containing transcendentals
or other functions in the coefficients of the Y'',Y', and Y terms at this
time nor will it work for non-homogeneous equations. In the latter case,
the user can, of course, get a partial solution by making the equation
homogeneous and solving with this program.
The program is still in some state of modification and any comments,
bugs, complaints etc. should be sent to ELL@MC. In addition, the author
would appreciate any info on successful or unsuccessful application of this
program.