# Proof : Adjoints

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

© BrainMass Inc. brainmass.com October 9, 2019, 5:52 pm ad1c9bdddfhttps://brainmass.com/math/matrices/proof-adjoints-sturm-liouville-theorem-70604

#### Solution Preview

Please see the attached file for the complete solution.

Thanks for using BrainMass.

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 ...

#### Solution Summary

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

$2.19