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1. Let's define the operator M as follows:
Mu = f(x) u'' + g(x) u' + h(x) u
Now define the adjoint of M as M* and let
M*v = (fv)'' - (gv)' + hv = fv'' + (2f' - g)v' + (f'' - g' + h)v

Show that (M*)* = M

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An adjoint proof is provided. The solution is detailed and well presented.

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1. Let's define the operator M as follows:
Mu = f(x) u'' + g(x) u' + h(x) u
Now define the adjoint of M as M* and let
M*v = (fv)'' - (gv)' + hv = fv'' + (2f' - g)v' + (f'' - g' + h)v

Show that (M*)* = M

Solution:

Given Mu = f(x) u'' + g(x) u' + h(x) u =f D2(u) + g D(u) + h (u).

Now define the adjoint of M as M* and
Let ...

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