The Solution Functor Sol_g ​ of an Inhomogeneous ODE

The Solution Functor ${\mathrm{Sol}}_g$$\operatorname{Sol}_g$ of an Inhomogeneous ODE

Let $A=\mathbb{C}[D]$$A = \mathbb{C}[D]$ and $V=C^{\mathrm{\infty }}(\mathbb{R},\mathbb{C})$$V = C^\infty(\mathbb{R},\mathbb{C})$.
Fix an operator $P(D)\in A$$P(D) \in A$ and an exponential‑polynomial function $g\in V$$g \in V$, with minimal annihilator $A_g(D)$$A_g(D)$.

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1. The Common Space $W$$W$

Both the unknown $y$$y$ and the right‑hand side $g$$g$ live naturally in the same finite‑dimensional subspace

$$W:={\ker }_V(A_{g}P).$$

Indeed, $A_g(D)g=0$$A_g(D)g = 0$ implies $g\in {\ker }_V(A_g)\subseteq W$$g \in \ker_V(A_g) \subseteq W$, and any solution $y$$y$ of $P(D)y=g$$P(D)y = g$ must satisfy $A_{g}Py=A_{g}g=0$$A_g P \, y = A_g g = 0$, so $y\in W$$y \in W$.

The space $W$$W$ is the value at $V$$V$ of the representable functor

$$F:=h^{A_{g}P}={\mathrm{Hom}}_A(A/(A_{g}P),-).$$

That is, $W\cong F(V)$$W \cong F(V)$.

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2. The Operator as an Endomorphism of $F$$F$

On $W$$W$, the operator $P(D)$$P(D)$ restricts to an endomorphism, because $A_{g}P(Pw)=P(A_{g}Pw)=0$$A_g P (P w) = P(A_g P w) = 0$. Functorially this restriction is the $V$$V$‑component of the natural transformation

$$\mu (P):F⟶F,\mu (P{)}_M(\psi )=\psi \circ (r\mapsto P\cdot r).$$

Under the identification $\psi \leftrightarrow \psi (\overline{1})$$\psi \leftrightarrow \psi(\bar 1)$, one has $\mu (P{)}_V(y)=P(D)y$$\mu(P)_V(y) = P(D) y$.

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3. The Right‑hand Side as a Point of $F(W)$$F(W)$

The function $g\in W$$g \in W$ defines an $A$$A$-module homomorphism

$${\phi }_g:A/(A_{g}P)⟶V,\overline{1}⟼g.$$

Since its image lies in $W$$W$, we may view ${\phi }_g$$\varphi_g$ as an element of $F(W)={\mathrm{Hom}}_A(A/(A_{g}P),W)$$F(W) = \operatorname{Hom}_A(A/(A_g P), W)$. Similarly, a candidate solution $y\in W$$y \in W$ corresponds to ${\phi }_y\in F(W)$$\varphi_y \in F(W)$ with ${\phi }_y(\overline{1})=y$$\varphi_y(\bar 1) = y$.

The equation $P(D)y=g$$P(D)y = g$ then becomes exactly

$$\begin{array}{}\text{(1)}& \mu (P{)}_W({\phi }_y)={\phi }_g\text{in }F(W).\end{array}$$

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4. Translation to Natural Transformations (Yoneda)

By the Yoneda lemma, an element of $F(W)$$F(W)$ is the same as a natural transformation ${\mathrm{Hom}}_A(W,-)\Rightarrow F$$\operatorname{Hom}_A(W,{-}) \Rightarrow F$.
Explicitly, to $\phi \in F(W)$$\varphi \in F(W)$ corresponds $\hat{\phi }$$\widehat{\varphi}$ with

$${\hat{\phi }}_M(\alpha )=\phi \circ \alpha ,\alpha \in {\mathrm{Hom}}_A(W,M).$$

Applying this to ${\phi }_y$$\varphi_y$ and ${\phi }_g$$\varphi_g$ gives

$$\widehat{{\phi }_y},\widehat{{\phi }_g}:{\mathrm{Hom}}_A(W,-)⟹F.$$

Because the Yoneda embedding is fully faithful, equation (1) is equivalent to the equality of natural transformations

$$\begin{array}{}\text{(2)}& \boxed{{\displaystyle \mu (P)\circ \widehat{{\phi }_y}=\widehat{{\phi }_g}}}.\end{array}$$

Thus the inhomogeneous ODE is precisely the statement that two natural transformations out of ${\mathrm{Hom}}_A(W,-)$$\operatorname{Hom}_A(W,{-})$ coincide.

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5. The Solution Functor as a Pullback

Equation (2) asks: which natural transformations $\widehat{{\phi }_y}$$\widehat{\varphi_y}$ satisfy it? This is exactly the description of a pullback in the functor category $[{\mathsf{Mod}}_A,\mathsf{Set}]$$[\mathsf{Mod}_A, \mathsf{Set}]$:

$$\begin{array}{}\text{(3)}& \begin{array}{ccc}{\mathrm{Sol}}_g& \stackrel{}{\to }& {\mathrm{Hom}}_A(W,-)\\ \downarrow & & \downarrow \widehat{{\phi }_g}& \\ F& \underset{\mu (P)}{\overset{}{\to }}& F\end{array}\end{array}$$

Concretely, for any $A$$A$-module $M$$M$,

$${\mathrm{Sol}}_g(M)=\{(\psi ,\alpha )\in F(M)\times {\mathrm{Hom}}_A(W,M)\mid \mu (P{)}_M(\psi )={\phi }_g\circ \alpha \}.$$

Setting $y=\psi (\overline{1})$$y = \psi(\bar 1)$, the condition reads

$$P(D)y=\alpha (g).$$

So an element of ${\mathrm{Sol}}_g(M)$$\operatorname{Sol}_g(M)$ is a family of solutions parametrised by a linear map $\alpha :W\to M$$\alpha : W \to M$ that transports the right‑hand side $g$$g$ into $M$$M$.

• The classical equation $P(D)y=g$$P(D)y = g$ is recovered by taking $M=V$$M = V$ and $\alpha =i:W\hookrightarrow V$$\alpha = i : W \hookrightarrow V$ (the inclusion).

• A general $\alpha$$\alpha$ yields a twisted equation $P(D)y=\alpha (g)$$P(D)y = \alpha(g)$.

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6. Representability and the Module $Q$$Q$

All three functors in diagram (3) are representable:

$$F={\mathrm{Hom}}_A(A/(A_{g}P),-),{\mathrm{Hom}}_A(W,-)\text{is represented by }W.$$

The natural transformations $\mu (P)$$\mu(P)$ and $\widehat{{\phi }_g}$$\widehat{\varphi_g}$ come from $A$$A$-module homomorphisms with a common source:

$$A/(A_{g}P)\stackrel{m_P}{\to }A/(A_{g}P),A/(A_{g}P)\stackrel{{\phi }_g}{\to }W,$$

where $m_P(\overline{f})=\overline{Pf}$$m_P(\bar f) = \overline{P f}$.

The Yoneda embedding sends colimits in ${\mathsf{Mod}}_A$$\mathsf{Mod}_A$ to limits in the functor category. The pullback (3) is therefore representable; its representing object is the pushout

$$Q:=(A/(A_{g}P)\oplus W)/⟨(m_P(a),-{\phi }_g(a))\mid a\in A/(A_{g}P)⟩.$$

We obtain

$${\mathrm{Sol}}_g\cong {\mathrm{Hom}}_A(Q,-).$$

Thus the module $Q$$Q$ encodes all equations $P(D)y=\alpha (g)$$P(D)y = \alpha(g)$ at once.
A single homomorphism $Q\to V$$Q \to V$ produces a solution to a specific equation; changing the homomorphism changes the equation and its solution coherently.

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7. Decomposition via the Chinese Remainder Theorem

Assume $A_{g}P$$A_g P$ splits completely over $\mathbb{C}$$\mathbb{C}$ (the general case is similar). Write

$$A_{g}P=\prod _{i=1}^r(D-{\lambda }_i{)}^{e_i+v_i},$$

where $e_i=v_{(D-{\lambda }_i)}(A_g)$$e_i = v_{(D-\lambda_i)}(A_g)$ and $v_i=v_{(D-{\lambda }_i)}(P)$$v_i = v_{(D-\lambda_i)}(P)$.
By the Chinese Remainder Theorem,

$$A/(A_{g}P)\cong \prod _{i=1}^{r}A/((D-{\lambda }_i{)}^{e_i+v_i}).$$

The subspace $W$$W$ decomposes into characteristic subspaces:

$$W=\underset{i=1}{\overset{r}{⨁}}{W}_i,W_i={\ker }_V((D-{\lambda }_i{)}^{e_i+v_i}).$$

The maps $m_P$$m_P$ and ${\phi }_g$$\varphi_g$ respect this decomposition.
Since finite direct sums commute with pushouts in ${\mathsf{Mod}}_A$$\mathsf{Mod}_A$, the representing module $Q$$Q$ splits as

$$Q\cong \underset{i=1}{\overset{r}{⨁}}{Q}_i,$$

where each $Q_i$$Q_i$ is the local pushout at ${\lambda }_i$$\lambda_i$:

$$Q_i=(A/((D-{\lambda }_i{)}^{e_i+v_i})\oplus W_i)/⟨(m_P^{(i)}(a),-{\phi }_{g,i}(a))⟩.$$

Correspondingly, the solution functor decomposes as a product:

$${\mathrm{Sol}}_g\cong \prod _{i=1}^r{\mathrm{Hom}}_A(Q_i,-).$$

This is why classical methods solve independently for each exponential term.