= Solution
The <space of test functions> is $\mathcal D(\mathbb R)=C_c^\infty(\mathbb R)$: its elements are <smooth functions> with <compact support>. For a fixed <compact set> $K$, put $\mathcal D_K=\{\varphi\in C^\infty(\mathbb R):\operatorname{supp}\varphi\subset K\}$ and use the <seminorms> $p_m(\varphi)=\max_{0\leq j\leq m}\|\varphi^{(j)}\|_\infty$. The <space of test functions> carries the usual <test-function inductive limit topology> of these spaces. In particular, \b[a sequence converges precisely when its supports eventually lie in one compact set and every <derivative> converges uniformly]. Thus $\varphi_j\to\varphi$ means that a common $K$ contains their supports and $p_m(\varphi_j-\varphi)\to0$ for every $m$.
A <distribution> is a <continuous linear functional> on this <space of test functions>, and $\mathcal D'(\mathbb R)$ denotes their space. We use complex-linear, bilinear pairings $\langle u,\varphi\rangle$. Equivalently, for every <compact set> $K$ there are $C_K$ and a nonnegative integer $m_K$ such that
$$
|\langle u,\varphi\rangle|\leq C_Kp_{m_K}(\varphi),\qquad \varphi\in\mathcal D_K.
$$
The usual <weak convergence of distributions> is $u_j\to u$ if $\langle u_j,\varphi\rangle\to\langle u,\varphi\rangle$ for every <test function>. This specifies the convergence used below; it does not require convergence in any norm.
The <distributional derivative> is defined by
$$
\boxed{\langle u',\varphi\rangle=-\langle u,\varphi'\rangle.}
$$
The <derivative> map sends $\mathcal D_K$ continuously to itself, with $p_m(\varphi')\leq p_{m+1}(\varphi)$. The preceding continuity estimate therefore proves that $u'$ is again a <distribution>. This definition extends the ordinary <derivative> of a <smooth function>, by <integration by parts>.
Choose the <translation of a distribution> convention $\tau_hf(x)=f(x-h)$. Its action on a <test function> is
$$
\langle\tau_hu,\varphi\rangle=\langle u,\varphi(\mathord\cdot+h)\rangle.
$$
For fixed $h$, the translated <test functions> have translated <compact support> and unchanged derivative sup norms, so this defines a <distribution>. For the <differentiability of distribution translations>, apply the <Taylor theorem> in integral form:
$$
\frac{\varphi(x-h)-\varphi(x)}h=-\int_0^1\varphi'(x-sh)\,ds\longrightarrow-\varphi'(x)
\quad\text{in }\mathcal D(\mathbb R).
$$
All supports lie in one slightly enlarged <compact set> for $|h|\leq1$, and the same identity for every <derivative> proves <uniform convergence>. The continuity of $u$ now gives
$$
\boxed{\lim_{h\to0}\frac{\tau_{-h}u-u}{h}=u'\quad\text{in }\mathcal D'(\mathbb R).}
$$
The minus sign in the translation parameter is necessary for this convention.
For (a), choose a <test function> $\eta$ with $\int\eta=1$. If $\int\psi=0$, then $F(x)=\int_{-\infty}^x\psi(s)\,ds$ is a <test function>, since the zero integral makes it vanish beyond both ends of the <compact support>. If $u_1'=0$, then $\langle u_1,\psi\rangle=\langle u_1,F'\rangle=0$. Decompose $\varphi=(\int\varphi)\eta+\psi$ to obtain
$$
\boxed{u_1=C\quad(C\in\mathbb C),\qquad \langle u_1,\varphi\rangle=C\int\varphi.}
$$
Conversely, these constant <regular distributions> have zero <distributional derivative>. This also proves the general fact that <a distribution with zero derivative is constant>.
For (b), choose a <cutoff function> $\rho$ equal to one near zero. Every <test function> decomposes as
$$
\varphi(x)=\varphi(0)\rho(x)+x\psi(x),\qquad
\psi(x)=\frac{\varphi(x)-\varphi(0)\rho(x)}x\in\mathcal D(\mathbb R).
$$
The quotient extends as a <smooth function> at zero, and it has <compact support>. If $xu_2=0$, the definition of <multiplication of a distribution by a smooth function> gives $\langle u_2,\varphi\rangle=\langle u_2,\rho\rangle\varphi(0)$. Conversely, $x\delta_0=0$. Thus the <kernel of multiplication by a coordinate> is exactly
$$
\boxed{u_2=C\delta_0.}
$$
In particular, derivatives of the <Dirac delta distribution> are not additional solutions: $x\delta_0'=-\delta_0$.
Now write $D=d/dx$ and $P(z)=-z^n+\sum_{j=0}^{n-1}a_jz^j$, where $n\geq1$. The <characteristic roots of a constant-coefficient differential equation> are the distinct <roots of a polynomial> $\lambda_1,\ldots,\lambda_r$ of $P$, with multiplicities $m_1,\ldots,m_r$. The <distributional regularity of a constant-coefficient ordinary differential equation> can be proved without assuming regularity in advance. Put $q_j(z)=(z-\lambda_j)^{m_j}$ and $Q=\prod_jq_j=-P$. Since the $q_j$ are <coprime polynomials>, polynomial division and <Bezout identity> give polynomials $b_j$ such that
$$
\sum_jb_j(z)\frac{Q(z)}{q_j(z)}=1.
$$
For example, invert $Q/q_j$ modulo $q_j$, sum the resulting expressions, and absorb a remaining multiple of $Q$ into one coefficient. For $P(D)v=0$, the <kernel decomposition for coprime polynomials> therefore gives
$$
v=\sum_jv_j,\qquad
v_j=b_j(D)\frac{Q(D)}{q_j(D)}v,\qquad
(D-\lambda_j)^{m_j}v_j=0.
$$
The <Leibniz rule> for <multiplication of a distribution by a smooth function> implies $D^{m_j}(e^{-\lambda_jx}v_j)=0$. Repeatedly using <a distribution with zero derivative is constant> shows that a <distribution> with $m$th derivative zero is a <polynomial> of degree at most $m-1$: subtract the polynomial primitive of its constant $(m-1)$st derivative, and induct on $m$. Consequently the most general <exponential polynomial solution of a constant-coefficient differential equation> is
$$
\boxed{v(x)=\sum_{j=1}^r e^{\lambda_jx}\sum_{\ell=0}^{m_j-1}c_{j\ell}x^\ell.}
$$
Every displayed term is annihilated by $P(D)$, so all coefficients are allowed. For the <linear independence> of these $n$ functions, on a relation, apply $\prod_{i\ne j}(D-\lambda_i)^{m_i}$; on $e^{\lambda_jx}$ times a polynomial of degree less than $m_j$, each remaining factor acts invertibly on that polynomial space. Hence the $j$th <polynomial> must vanish. \b[Every distributional solution is an <analytic function>, and in particular a <classical solution>.] For real coefficients and real-valued <distributions>, take <complex conjugate> coefficients at conjugate <characteristic roots of a constant-coefficient differential equation>, or equivalently use the corresponding real sine and cosine forms.
For the last equation, the operator is $xP(D)$: the coordinate multiplies the result of differentiation. The <kernel of multiplication by a coordinate> says precisely that
$$
xP(D)u=0\quad\Longleftrightarrow\quad P(D)u=C\delta_0.
$$
Choose the <retarded fundamental solution of a constant-coefficient ordinary differential operator> $E=Hw$, where $H$ is the <Heaviside function> and $w$ is the analytic solution of $P(D)w=0$ with
$$
w^{(j)}(0)=0\quad(0\leq j\leq n-2),\qquad w^{(n-1)}(0)=-1.
$$
Such $w$ exists uniquely by the elementary <initial value problem> for a constant-coefficient <ordinary differential equation>; alternatively the residue construction in the next solution gives it explicitly. The <distributional jump formula for a Heaviside product> is
$$
D^k(Hw)=Hw^{(k)}+\sum_{s=0}^{k-1}w^{(k-1-s)}(0)\delta_0^{(s)}.
$$
It follows by induction from $D(Hw)=Hw'+w(0)\delta_0$, using <integration by parts>. With the chosen initial derivatives, every lower-order jump term vanishes and the leading coefficient $-1$ gives $P(D)E=\delta_0$. Subtracting $CE$ reduces the last equation to the homogeneous one. Thus the <coordinate-degenerate constant-coefficient differential equation> has exactly the solutions
$$
\boxed{u(x)=\sum_{j=1}^r e^{\lambda_jx}\sum_{\ell=0}^{m_j-1}c_{j\ell}x^\ell+C\,H(x)w(x).}
$$
There are $n+1$ independent constants. For $n\geq2$, derivatives through order $n-2$ match at zero, while the derivative of order $n-1$ may jump; for $n=1$, the function itself may jump. An arbitrary <distribution> concentrated at zero cannot be added, because its image under the nonzero leading derivative would contain a nonvanishing highest derivative of the <Dirac delta distribution>.
Back to article page
