To insure that these solutions be independent, take
Then the Wronskian at
which implies the linear independence of and on I.
You start by plugging the power series (23) into
equation (22). Try to write the resulting equation as a single series set equal
to zero. From this you should
be able to extract a recurrence relation for the coefficients
. Then use the
recurrence relation and (23) solve for the . Use the same recipe to de termine the
. A few points:
a. The method of series solutions is only practical in a few cases where the
are very simple . As we saw in class, the method worked on the Airy equation
u''− tu = 0,
b. When solving an initial value problem , you should choose the point
and (22) to be one at which
the initial values are given.
c. The power series might only converge in a small interval centered at
5. A Cauchy-Euler (equidimensional) equation is one of the form
For an equation of this sort, you can always find a
fundamtental set. Look for solutions of the form u = tm.
Plug this into (28) to obtain