# Are adjoint operators historically related to integrating factors?

Birkhoff and Rota, in their book Ordinary Differential Equations (4e), claim on p.55 that:

The concept of the adjoint of a linear operator, which originated historically in the search for integrating factors, is of major importance because of the role which it plays in the theory of orthogonal and biorthogonal expansions.

I am interested in substantiations of the historical claim of Birkhoff and Rota; i.e. the claim that the roots of the notion of an adjoint is in the pursuit of finding integrating factors. It is admissible to interpret the word "adjoint" with arbitrarily specific scope. Different words signifying similar ideas are also admissible. (Consequently my question is not about terminology; compare What is the first usage of the term "Adjoint" and why was this word chosen?, The terminologies "Adjoint" and "Adjugate")

For context, this is how Birkhoff and Rota define the adjoint: for $$L= p_2 D^2+p_1 D+p_0$$ a linear differential operator,

• $$L$$ is exact if for some $$q_1,q_0$$, $$L = D(q_1 D + q_0)$$,
• Thus $$L$$ is exact iff $$p_2''-p_1'+p_0=0$$,
• $$v$$ is an integrating factor for $$L$$ if $$vL$$ is exact,
• Thus $$v$$ is an integrating factor iff $$M[v]=(vp_2)''-(vp_1)'+(vp_0) = 0$$,
• $$M = p_2D^2 + (2p_2'-p_1)D + (p_2''-p_1'+p_0)$$ is the adjoint of $$L$$.

(Thus for some appropriately chosen functions, $$\langle v,L[u]\rangle = \langle M[v], u\rangle$$, the bilinear pairing being through an integral.)