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Let $h_1,\dots ,h_n$ be a basis of homogeneous solutions of a linear ordinary differential operator $L$ of order $n$. Let $B_1,\dots ,B_n$ be independent homogeneous linear boundary conditions, which may be coupled. Suppose a fundamental solution $G_0(x,\xi )$ satisfies $L_x{G}_0=\delta (x-\xi )$. Define the boundary matrix $M_{ij}=B_i{h}_j$ and $b_i(\xi )=B_i{G}_0(\cdot ,\xi )$. If $M$ is invertible, thenThe added terms are homogeneous solutions, so the Dirac delta source is preserved. Applying the boundary operators gives $b-M{M}^{-1}b=0$. Any two such Green functions differ, for fixed source position, by a homogeneous solution with coefficient vector in the null space of $M$; invertibility therefore proves uniqueness. A singular boundary matrix signals a nontrivial homogeneous solution satisfying the endpoint conditions, where a compatibility condition and a nonunique inverse may arise. For $L=D^2$, one convenient fundamental solution is $(x-\xi )H(x-\xi )$.

For a real linear differential operator $L$, one has $L(\mathrm{Im}u)=\mathrm{Im}(Lu)$. A forcing term written as the imaginary part of a polynomial times $e^{i\omega x}$ can therefore be handled by solving one complex equation, then taking the imaginary part. For $L=d^2/d{x}^2+a$, substituting $u=p(x)e^{i\omega x}$ gives $p^{\prime \prime }+2i\omega p^{\prime }+(a-{\omega }^2)p$. If the forcing frequency is resonant, the polynomial ansatz must include the appropriate additional power of $x$.

If a differential operator $L$ is a linear map and has nonzero kernel of a linear map, the equation $LA=J$ determines $A$ only up to a homogeneous solution. Lorentz covariance of $L$ and $J$ therefore does not force independently selected solutions in different inertial frames to obey a particular tensor transformation law. A covariant Green function and consistently transformed homogeneous data do provide such a choice. For the four-potential in Lorenz gauge, $\square A^{\mu }=-{\mu }_0{J}^{\mu }$ still permits $A^{\mu }\mapsto A^{\mu }+{\partial }^{\mu }\chi$ with $\square \chi =0$, a residual Lorenz gauge symmetry. Constant spatial potentials for zero sources are a concrete example.

A differential operator is in divergence form when its highest derivatives occur inside a divergence, for example $Lu=\nabla \cdot (A\nabla u)$ or $(p(x)u^{\prime }{)}^{\prime }$. This form makes integration by parts express the operator through a lower-order bilinear form and boundary flux.

The generalized Leibniz rule extends composition of differential operators to all integer $n$. Define $\left(\genfrac{}{}{0ex}{}{n}{r}\right)=n(n-1)\cdots (n-r+1)/r!$. For nonnegative $n$ the sum stops at $r=n$. For $n=-s<0$, its coefficients are $(-1{)}^r(\genfrac{}{}{0ex}{}{s+r-1}{r})$; in particular ${\partial }^{-1}f=f{\partial }^{-1}-f^{\prime }{\partial }^{-2}+f^{\prime \prime }{\partial }^{-3}-\cdots$. Applying $\partial$ on the left cancels consecutive derivative terms and recovers multiplication by $f$, which explains the inverse expansion.

A Fourier mode is a single complex oscillatory component $e^{i\omega x}$. Its real and imaginary parts are a cosine and a sine; $\omega$ is the spatial angular frequency. Every constant-coefficient differential operator acts on it by multiplication: $P({\partial }_x)e^{i\omega x}=P(i\omega )e^{i\omega x}$. In particular, ${\partial }_x^2{e}^{i\omega x}=-{\omega }^2{e}^{i\omega x}$. These eigenfunctions reduce a partial differential equation such as the heat equation to separate equations for each frequency. On the real line a pure Fourier mode is not in the L2 space; the Fourier transform uses such modes in a continuous superposition.

For a fixed formal pseudodifferential operator $P={\sum }_j{p}_j{\partial }^j$ and a monic differential operator $L={\partial }^m+{\sum }_{l=0}^{m-1}{u}_l{\partial }^l$, cyclicity of the Adler trace gives $l_P(L)=\mathrm{Tr}(PL)$ with $c=\int p_{-m-1}dx$ and $a_l=\mathrm{res}({\partial }^{l}P)={\sum }_{k=0}^l\left(\genfrac{}{}{0ex}{}{l}{k}\right)p_{k-l-1}^{(k)}$. The function is affine in the variable coefficients; its variational derivative with respect to $u_l$ is $a_l$.

For the sign convention $\Delta =\mathrm{div}\nabla$, the heat operator is the differential operator $L={\partial }_t-\Delta$. The heat equation is $Lu=0$. If $P=-\Delta$ is the positive Laplace-Beltrami operator, the same operator is ${\partial }_t+P$. A Riemannian heat kernel is a fundamental solution with the initial Dirac delta distribution.

A homogeneous linear differential equation is a linear differential equation with identically zero forcing term, $L[y]=0$. On a common interval, its solutions of a differential equation form the vector space $\ker L$, the kernel of a linear map. They include the zero function and are closed under addition and scalar multiplication. For any two solutions of a differential equation $u,v$ and constant scalars $\alpha ,\beta$, linearity gives $L[\alpha u+\beta v]=\alpha 0+\beta 0=0$: this is the superposition principle. The scalars must be constant; variable multipliers introduce extra derivatives.

For example, $y^{\prime }+2y=0$ has general solution $y=C{e}^{-2t}$, where $C$ is any constant real number. A short verification uses the product rule: $(e^{2t}y{)}^{\prime }=e^{2t}(y^{\prime }+2y)=0$, so $e^{2t}y$ is constant. For $y_1=e^{-2t}$ and $y_2=2e^{-2t}$, the sum is $3e^{-2t}$ and also solves the same homogeneous linear differential equation. At $t=1$, their values are approximately $0.13534$, $0.27067$ and $0.40601$. Every curve is a multiple of one exponential function; the solution space of a homogeneous linear differential equation has dimension one. Both $t$ and $y$ in this example and the following diagram are dimensionless mathematical variables.

Now keep the same differential operator $L[y]=y^{\prime }+2y$ but prescribe the forcing term $1$. The inhomogeneous linear differential equation $y^{\prime }+2y=1$ has particular solution $y_p=1/2$ and general solution $y=1/2+C{e}^{-2t}$. The initial conditions $y(0)=0,1,2$ select, respectively,At $t=1$, their values are approximately $0.43233$, $0.56767$ and $0.70300$. The midpoint $\frac{1}{2}(u_0+u_2)=u_1$ solves the same inhomogeneous linear differential equation, because $L[\frac{1}{2}(u_0+u_2)]=\frac{1}{2}(1+1)=1$. The sum $s=u_0+u_2=1+e^{-2t}$ instead satisfies $s^{\prime }+2s=2$, so it fails the equation with forcing term $1$. More generally, a linear combination of solutions of a differential equation for the same fixed nonzero forcing term retains that forcing term precisely when its scalar weights add to one. This is the affine space structure, rather than unrestricted superposition within that fixed family.

On the left, all three curves solve the same homogeneous linear differential equation. On the right, $u_0,u_1,u_2$ share the particular solution $1/2$, and the shaded difference $u_2-1/2$ is the homogeneous solution $\frac{3}{2}{e}^{-2t}$. The dashed sum has twice the forcing term, even though its starting value coincides with that of $u_2$. Adding the two initial conditions also adds their inputs; taking their midpoint averages both. This distinction makes the superposition principle useful for assembling responses to different inputs without mistaking that assembly for a solution of a differential equation with the original input. See MIT's superposition notes, including the particular solution one half.

An integral kernel is a function or distribution specifying an integral operator through $(Tf)(x)=\int K(x,y)f(y)dy$. A Green function is an integral kernel that inverts a differential operator under specified boundary conditions.

Two Riemannian manifolds are isospectral for a specified differential operator if their eigenvalues, counted with multiplicities and with the same boundary conditions, agree. Riemannian isometries preserve the Laplace-Beltrami operator spectrum, but the converse can fail. The transplantation theorem and the Sunada theorem give systematic constructions.

The Jacobi identity is the compatibility condition on a Lie bracketTogether with an Alternating Lie bracket, it defines a Lie algebra. Unlike associativity, it relates three cyclic bracketings rather than equating two bracketings. Equivalently, using antisymmetry of a Lie bracket,Thus the linear map ${\mathrm{ad}}_x:y\mapsto [x,y]$ is a derivation of a Lie algebra: acting on a bracket is the sum of acting on each argument. This makes infinitesimal symmetries act consistently on each other's products. It also gives $[{\mathrm{ad}}_x,{\mathrm{ad}}_y]={\mathrm{ad}}_{[x,y]}$, the identity that makes the Adjoint representation of a Lie algebra a Lie algebra representation.

For the commutator in an associative algebra, the cancellation is directly visible:Each word occurs once with each sign. This proves the Jacobi identity without a restriction on the characteristic of a field.

For a computed check in the sl2 matrix commutator example,Here the last two terms are individually nonzero over characteristic zero; the identity records their cancellation. In the cross-product Lie algebra, the same cyclic cancellation is the vector triple product identity.

The historical connection is unusually concrete: Lie and Engel's volume I expands nested commutators of differential operators into twelve terms and cancels them cyclically, in Chapter 5, §26, equation (7). See Lie and Engel, §26, in Merker's translation, pp. 105–106.

The differential operator $L=\frac{1}{2}{x}^2{\partial }_{xx}+\frac{1}{2}x{\partial }_x-\frac{1}{2}$ has solution operator $P_{t}g(x)=e^{-t/2}\mathbb{E}[g(x{e}^{B_t})]$. For $x\ne 0$ its integral kernel with respect to Lebesgue measure is $p(t,x,y)=e^{-t/2}(|y|\sqrt{2\pi t}{)}^{-1}\exp [-{\log }^2(|y/x|)/(2t)]$ when $xy>0$, and zero otherwise. For $x=0$ the kernel is the measure $e^{-t/2}{\delta }_0$, not a density. The Itô formula applied to $e^{-s/2}u(t-s,X_s)$ proves this Feynman-Kac formula.

A Lie algebra over a field $k$ is a vector space $\mathfrak{g}$ with a bilinear map $[,]:\mathfrak{g}\times \mathfrak{g}\to \mathfrak{g}$, called the Lie bracket, that is alternating and satisfies the Jacobi identity. Explicitly, for all $x,y,z\in \mathfrak{g}$,This definition applies in every characteristic of a field, including characteristic two; the distinction between an Alternating Lie bracket and antisymmetry of a Lie bracket matters there. See Pavel Etingof, Lie Groups and Lie Algebras, Definition 8.4 and Remark 8.5.

The motivating question is how to describe very small symmetries. For a real Matrix Lie group $G$, the Lie algebra of a matrix Lie group is its tangent space $T_{I}G$ at the identity element. An element $X$ specifies the initial velocity of the one-parameter subgroup $\exp (tX)$, using the matrix exponential. The Lie bracket records the first nonzero effect of changing the order of two small transformations. Even when first-order motions cancel, a second-order motion can remain: the small rotation commutator below leaves a rotation about a third axis.

This turns local symmetry questions into computations with linear maps. A basis and its structure constants of a Lie algebra determine every Lie bracket by bilinearity. For example, three matrices suffice for the sl2 matrix commutator example, while the familiar cross product describes the SO(3) Lie algebra. These descriptions help calculate rotations in three dimensions and build Lie algebra representations; the same bracket structure underlies angular momentum commutation relations. The striking feature is that a nonlinear family of motions has a local description by a vector space and one product. Global information still needs care: the Lie algebra alone does not distinguish the circle group from the additive real line.

Historically, the connection with differential equations is visible in Sophus Lie's Theorie der Transformationsgruppen, volume I (1888), prepared with Friedrich Engel. Chapter 6 interprets first-order differential operators as infinitesimal transformations, linking their action on functions to the transformations those functions admit. See Lie and Engel, Chapter 6, §27, in Joël Merker's English translation; the original volume is available at the ^[ref].

A scalar linear differential equation of order $n\ge 1$ has the formHere the coefficients $a_j$ and the forcing term $g$ are prescribed functions of the independent variable $t$. A classical solution of a differential equation is an $n$-times differentiable function satisfying the identity at every point of its interval. The unknown function $y$ and its derivatives enter to the first power, without products between them or coefficients depending on $y$. Equivalently, the differential operator $L$ is a linear map: for constant scalars $\alpha ,\beta$,The forcing term may be zero, giving a homogeneous linear differential equation, or nonzero, giving an inhomogeneous linear differential equation. Linearity of $L$ does not mean that every function graph of a solution of a differential equation is a straight line.

To use the usual order-$n$ initial value problem theory on an interval $I$, assume that $a_n(t)\ne 0$ throughout $I$ and that all $a_j$ and $g$ are continuous functions there. Dividing by $a_n$ normalizes the highest derivative; specifying $y(t_0),y^{\prime }(t_0),\dots ,y^{(n-1)}(t_0)$ at $t_0\in I$ then selects one solution of a differential equation throughout $I$. A zero of $a_n$ needs separate analysis: one cannot divide through it and retain these conclusions. The first-order normalization appears in OpenStax's definition and standard form of linear equations; the corresponding system theorem is in MIT's existence and uniqueness notes for linear systems.

The coefficients need not be linear functions of $t$. For example, $t^2{y}^{\prime }+y=0$ is a linear differential equation on $t>0$, although $t^2$ is quadratic. Its solution of a differential equation with initial condition $y(1)=1$ is $y(t)=e^{1/t-1}$: its derivative is $y^{\prime }=-y/t^2$, so substitution gives $t^2{y}^{\prime }+y=0$, and $y(2)=e^{-1/2}\approx 0.60653$. The zero of the highest coefficient at $t=0$ explains the domain restriction. In contrast, $y^{\prime }+y^2=0$, $y{y}^{\prime }+y=0$ and $y^{\prime }+\sin y=0$ are nonlinear ordinary differential equations because of their dependence on the unknown function. Variable coefficients and nonlinear ordinary differential equations are different issues; see MIT's definitions and examples of first-order linear equations.

The practical benefit of linearity is that it separates the effect of the initial condition from the effect of the forcing term. Once one particular solution $y_p$ satisfies $L[y_p]=g$, every solution of a differential equation isHere $\ker L$ is the kernel of a linear map, consisting of solutions of a differential equation to the associated homogeneous linear differential equation. Indeed, $L[y-y_p]=g-g=0$, and conversely $L[y_p+h]=g+0=g$. Thus the family for a fixed forcing term is an affine space obtained by translating $\ker L$. If $g$ is not identically zero, the zero function is absent, so this family is not a vector space under the usual operations on functions. This short argument explains why a general solution is a particular solution plus a homogeneous solution.

A linear differential operator is a finite linear combination of derivatives with specified coefficient functions. It obeys $L(\alpha f+\beta g)=\alpha Lf+\beta Lg$ for scalar constants $\alpha ,\beta$. Variable coefficients need not commute with differentiation: $D(af)=a^{\prime }f+a{f}^{\prime }$.

A Neumann eigenfunction is an eigenfunction of a spatial differential operator whose normal derivative vanishes on the boundary. A Sturm-Liouville eigenfunction expansion in these functions gives a heat kernel for homogeneous Neumann boundary conditions.

For $U_x=iL(x)U$ with a self-adjoint transverse differential operator $L(x)$, differentiating the squared L2 norm gives $i⟨U,LU⟩-i⟨LU,U⟩=0$. Reality of the refractive coefficient and vanishing transverse boundary flux are required. This is exact for the scalar envelope model. Its physical interpretation as leading paraxial power uses a fixed reference impedance; a varying local impedance or a longitudinal electromagnetic polarization requires a consistent Maxwell calculation and cannot be inferred from this invariant alone.

The principal symbol of an odd-order scalar differential operator is an odd map from the frequency sphere to the complex plane. The Borsuk-Ulam theorem forces a zero on a two-dimensional sphere, contradicting ellipticity. Restriction to three variables proves the same obstruction in every greater dimension.

For the relative differential operators over $\mathbb{C}$ on a polynomial ring, coordinates have order zero and partial derivatives have order one. The associated graded ring is $\mathbb{C}[x,\xi ]$ with degrees zero for $x_i$ and one for ${\xi }_i$. This differs from the Bernstein filtration, in which both sets of generators have degree one and every filtration piece is finite-dimensional.