Simple expansion will show that

Thus,
Provided that there is no interval

on which

.
If there is such an interval for some

, then the operator

is not truly invertible; we have:
![\left(1-D^2+D^4-D^6+...\right)\left[\left(1+D^2\right)f(x)\right] = 0](latexrender/pictures/cadb7b3078c5434dc75a9bf93c242880.png)
on this interval.
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Let us suppose that

is a linear operator, and that

is the most general solution of the equation

(

is in general a parameterized family of solutions

). Then

Applying the (partial) inverse

to the right-hand equation then yields the most general possible solution:

In the particular case we are considering,

, the general solution to

is

Which leads directly to the same answer obtained in Exercise [21.02].