 
Summary: On d'Alembert substitution
S.A.Abramov
Computer Center of the Russian Academy of Science
Vavilova 40, Moscow 117967, Russia
email: abramov@sms.ccas.msk.su
Abstract
Let some homogeneous linear ordinary di#erential equation with coef
ficients in a di#erential field F be given. If we know a nonzero solution #,
then the order of the equation can be reduced by d'Alembert substitution
y = #
# v dx , where v is a new unknown function. In the situation when
# # F , after d'Alembert substitution an equation with coe#cients in F
arises again. Let the obtained equation have a nonzero solution # # F ,
then it is possible to reduce the order of the equation again and so on,
until an equation without nonzero solutions in F is obtained.
If we can find solutions not only in F but in some larger set L as
well (L can be a field or a linear space), then we can build up a certain
subspace M (d'Alembertian subspace) of the space of all solutions of the
original equation. Thus if we have algorithms AF and AL to search for the
solutions in F and L, then by incorporating d'Alembert substitution we
