read desing:  # read the Maple code (for Maple 8 and up).

with(DEtools):
_Envdiffopdomain := [Dx,x];

# An example with only one singularity, which is an apparant singularity.
lprint("Example 1:");
L := LCLM(Dx^2-x,Dx^2-x-x^2);
Multiply_on_the_left_by := desing(L);
DesingularizedOperator := mult(%, L);  # finite singularities are gone.


# Some more example where all finite singularities are apparant.
#
lprint("Example 2:");
L := LCLM(Dx^3-x^2,Dx^4-x^2);
Multiply_on_the_left_by := desing(L);
DesingularizedOperator := mult(%, L);  # finite singularities are gone.

lprint("Example 3:");
L := LCLM( Dx^3+x, Dx^4-x);
Multiply_on_the_left_by := desing(L);
DesingularizedOperator := mult(%, L);  # finite singularities are gone.


lprint("Example 4:");
# This desing implementation always finds a desingularization of minimal
# order but is not optimal in the sense that it usually
# makes the singularity at x=infinity worse even in the special
# case where this is avoidable. For example
L := LCLM(Dx-1/(x-1), Dx-3/x);
# has a desingularization Dx^4 that is simpler than
# the desingularization of the same order found by our implementation:
Multiply_on_the_left_by := desing(L);
DesingularizedOperator := mult(%,L);


lprint("Example 5:");
# An example in the paper:
L := Dx^2+1: # has solutions cos(x),sin(x).
L := symmetric_product(L, Dx-1/x); # has solutions x*cos(x), x*sin(x).
Multiply_on_the_left_by := desing(L);
DesingularizedOperator := mult(%,L);