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:

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].