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For the real-coefficient differential operator $Mv=(p{v}^{\prime }{)}^{\prime }+qv$, integration by parts givesIf $p$ tends to zero at both endpoints and $u,v,u^{\prime },v^{\prime }$ remain bounded there, the boundary term vanishes without prescribing endpoint values. Thus regular endpoint behaviour can supply the domain condition that makes the expression symmetric. Endpoint degeneracy alone is insufficient for arbitrary singular functions: for $p=x(1-x)$ and $v=\log x$, one has $p{v}^{\prime }=1-x\to 1$ at zero. A self-adjoint differential operator requires a specified admissible domain, rather than only a formal differential expression.

Variation of parameters replaces the constants in a complementary solution by functions to construct a particular solution.

For the inhomogeneous linear differential equation $y^{\prime \prime }+p{y}^{\prime }+qy=r$, assume $p,q,r$ are continuous functions on an interval $I$, and $y_1,y_2$ are linearly independent homogeneous solutions there. With $W=W(y_1,y_2)$ and $x_0\in I$, variation of parameters constructs the particular solutionThe Wronskian has no zeros on $I$, so these quotients are defined. For $A{y}^{\prime \prime }+B{y}^{\prime }+Cy=F$, normalize first: $r=F/A$, not $F$.^[ref]

To derive the formula, write $y_p=u_1{y}_1+u_2{y}_2$ and impose $u_1^{\prime }{y}_1+u_2^{\prime }{y}_2=0$. This auxiliary choice removes extra terms from the first derivative. Substitution into the inhomogeneous linear differential equation leaves $u_1^{\prime }{y}_1^{\prime }+u_2^{\prime }{y}_2^{\prime }=r$. The two linear equations for $u_1^{\prime },u_2^{\prime }$ have determinant $W$, and Cramer's rule givesIntegrating from $x_0$ makes $y_p(x_0)=y_p^{\prime }(x_0)=0$. Arbitrary integration constants contribute only homogeneous solutions; adding $c_1{y}_1+c_2{y}_2$ supplies the general solution.

For a worked example, use the same homogeneous solutions $e^{-x},e^{2x}$ and the forcing term $r=6e^{3x}$ in $y^{\prime \prime }-y^{\prime }-2y=6e^{3x}$. With $x_0=0$ and $W=3e^x$,Consequently the particular solution selected by zero initial conditions isIts value and first derivative vanish at zero. Direct substitution confirms the forcing term: the differential operator $L[y]=y^{\prime \prime }-y^{\prime }-2y$ annihilates the last two terms and maps $e^{3x}$ to $4e^{3x}$. The two integrals therefore build a response from known homogeneous solutions, even when the forcing term has no convenient elementary guess.

Author of teaching texts on partial differential equations and asymptotic analysis. These treatments connect differential operators, boundary conditions and approximation methods for equations of mathematical physics.^[ref]

On $(0,L)$, the differential operator $\mathcal{L}={\partial }_x^2+\beta {\partial }_x=e^{-\beta x}{\partial }_x(e^{\beta x}{\partial }_x)$ with homogeneous Neumann boundary conditions is self-adjoint in the weighted inner product with weight $e^{\beta x}$. Its eigenfunctions and nonnegative decay rates areTheir squared norms are $N_0=(e^{\beta L}-1)/\beta$ and $N_k=L[1+{\beta }^2/(4q_k^2)]/2$. At $\beta =0$, interpret the continuous limit as $N_0=L$. The Sturm-Liouville eigenfunction expansion givesThis kernel acts against the measure $e^{\beta y}dy$, not Lebesgue measure alone. It is symmetric in $x,y$; the transition density against Lebesgue measure is $e^{\beta y}{K}_{\beta }(x,y,t)$.

The first Weyl algebra over a field $k$ isIn characteristic zero, it acts faithfully on $k[x]$: $x$ multiplies by the variable and $\partial$ differentiates. The defining relation is the product rule, since $\partial (xf)-x\partial (f)=f$. Every element has a unique ordered expression ${\sum }_{i,j}{a}_{ij}{x}^i{\partial }^j$, giving the ordered monomial basis of a Weyl algebra.

The higher Weyl algebra $A_n(k)$ has commuting coordinates $x_i$, commuting derivatives ${\partial }_i$, and $[{\partial }_i,x_j]={\delta }_{ij}$. It organizes polynomial-coefficient differential operators. The abstract algebra remains meaningful in positive characteristic, but its usual action on polynomials is no longer faithful: ${\partial }^p$ acts as zero in characteristic $p$ while remaining a nonzero ordered monomial. See Weyl algebra in positive characteristic for the resulting finite-dimensional representations.

Over $\mathbb{C}$, the Witt algebra is the Lie algebra of algebraic derivations of the Laurent polynomial ring $\mathbb{C}[z,z^{-1}]$. It has basis ${\ell }_n=-z^{n+1}d/dz$ for $n\in \mathbb{Z}$. The commutator of these differential operators isIndeed, the second-derivative terms cancel in the commutator, leaving the derivative of the coefficient of each operator. The span of ${\ell }_{-1},{\ell }_0,{\ell }_1$ is a copy of the sl2 Lie algebra: take $e={\ell }_{-1}$, $h=2{\ell }_0$, $f=-{\ell }_1$.

The Witt algebra models finite Fourier series of complexified vector fields on the circle. Its nontrivial one-dimensional central extension of a Lie algebra is the Virasoro algebra, whose central term changes the representation theory relevant to conformal field theory.