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Exercise [21.03]
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Joined: 12 Jul 2010, 07:44
Posts: 154
Exercise [21.03]
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