Rit (Vísindafélag Íslendinga) - 01.06.1951, Page 105
105
[5]
In stead of (34) we may apply (34b) or (34d) to obtain
the solution. I abstain from writing these solutions.
3.
whence we obtain the solution as the basal function
In spite of different aspect the solutions 2. and 3. are identical
as will be evident from the fact that their serial relations or
derivatives at zero are identical. On the other hand they are,
in a different way, illuminating for the function they repre-
sent. From solution 2. we see at once that y is a polynomial
ax2
in x multiplied by e 2 if b—{2m-\-\)a, m being a posi-
tive initeger or zero, while solution 3. shows that y is a
polynomial in x2 if b=—2ma. Solution 3. is to some extent
more flexible as we are able to expand it into various series.
Substituting solution 2. in the original differential equation
it is convenient to make use of the following rule: —
xÐpx(xqv) = ÐP(xt’+Iv)—pÐP-,(x'iv) |
xzÐp(xqv) =Ðp(xq+21;)- 2/^-'(jc<'+/ v)+p(p~\ )Ðp~2(xp v) (66)
etc.
which we derive readily from
Ðpx(x • jc'»v) = xÐp(xqv) + pÐP-^x* v)
Solution 3. satisfies also the original equation. It is, how-
ever, not out of the way to indicate, in this case, how we
shall ascertain that a basal function is zero, identically. After
having introduced the basai function (Solution 3.) into the dif-
ferential equation and carried out the indicated differentiations
we arrive, after some obvious simplifcations, at the following:—