Equations in distributions

Prove that the generalized function $f = \sum_{k = 0}^{n-1} c_{k} \delta_{0}^{(k)}, c_{0}, \ldots , c_{n-1} \in \mathbb {C}$ is the general solution in $\mathcal{D}^{\prime }(\mathbb {R})$ of the equation $x^{n} f(x) = 0, n \in \mathbb {N}$.

Let $f \in \mathcal{D}^{\prime }(\mathbb {R})$ and $f(x+a) = f(x)$ for all $a \in \mathbb {R}$. Prove that $f =$ const, i.e., there exists $C \in \mathbb {C}$ such that $\left\langle f, \varphi \right\rangle = C \int_{\mathbb {R}} \varphi (x) d x, \varphi \in \mathcal{D}(\mathbb {R})$.

Find the general solution of the equation $(x+1) y^{\prime \prime } = 0$ in $\mathcal{D}^{\prime }(\mathbb {R})$.

Let $a \in \mathbb {R} \backslash \left\{ 0\right\} , \varphi \in \mathcal{D}(\mathbb {R})$. Find the general solution of the equation $\frac{d^{2} u}{d x^{2}}+ a^{2} u = \varphi$.

Let $\alpha \in C^{\infty }(\mathbb {R})$ have a unique zero of order 1 at the point $x = 0$. Find the general solution of the equation $\alpha f = 0$ in $\mathcal{D}^{\prime }(\mathbb {R})$.

Find the general solution of the equation $y^{\prime \prime } = 0$ in $\mathcal{D}^{\prime }(\mathbb {R})$.

Prove that the general solution of the equation $x u^{\prime } = 1$ in $\mathcal{D}^{\prime }(\mathbb {R})$ has the form $u = c_{1}+c_{2} \theta (x)+\ln \left|x\right|$, where $c_{1}, c_{2} \in \mathbb {C}$ are arbitrary and $\theta$ is the Heaviside function.

Find the general solutions in $\mathcal{D}^{\prime }(\mathbb {R})$ of the following equations

Find the general solution in $\mathcal{D}^{\prime }(\mathbb {R})$ of the equations

Prove that for $n > m$ the general solution in $\mathcal{D}^{\prime }(\mathbb {R})$ of the equation $x^{n} u^{(m)} =$ 0 is the generalized function

where $a_{k}, b_{k}$ are arbitrary complex numbers.

Find the fundamental solution of the operator $D+a I, a \in \mathbb {R}$, and the general solution of the equation $\frac{d u}{d x}+a u = \varphi , \varphi \in \mathcal{D}(\mathbb {R})$.

Find the fundamental solution of the operator $L$ and the general solution of the equation $L u = \varphi , \varphi \in \mathcal{D}(\mathbb {R})$.

Prove that the generalized function

$\varphi \in \mathcal{D}\left(\mathbb {R}^{2}\right)$, satisfies the equation $\left(x^{2}+y^{2}\right) f = 1$ in $\mathcal{D}^{\prime }\left(\mathbb {R}^{2}\right)$.

Solve in $\mathcal{D}^{\prime }$ the equations:

Prove that every generalized function $G$ has an antiderivative, that is, a generalized function $F$ such that $F^{\prime } = G$. Consider the cases:

Solve in $\mathcal{D}^{\prime }$ the differential equations:

Prove that a generalized function $F$ is invariant with respect to shifts ($\langle F, \varphi (x)\rangle = \langle F, \varphi (x+a)\rangle$ for all $\varphi \in \mathcal{D}$ and $a \in \mathbb {R})$ if and only if it coincides with the regular function generated by some constant.

Prove that any generalized function satisfying the equation

with constant coefficients $a_{k} \in \mathbb {C}$ is a regular generalized function, generated by some ordinary solution of this equation.

Find all functions $f \in \mathcal{S}^{\prime }$ satisfying the equation:

Prove that a generalized function $F\in \mathcal D'(\mathbb {R})$ with the property that $F'=0$ is a constant.

Prove that all the solutions of the equation $xF=0$ among the generalized functions $F\in \mathcal D'(\mathbb {R})$ are proportional to the $\delta$-function.

A generalized function $F$ on the line is said to be homogeneous of degree $(\lambda ,\varepsilon )$, where $\lambda \in \mathbb {C}$ and $\varepsilon =0$ or $1$, if $F(tx)=\left|t\right|^{\lambda }(\operatorname {sgn}t)^\varepsilon F(x)$ for $t\ne 0$.

Prove that a homogeneous function of degree $(\lambda ,\varepsilon )$ on the line (see first-edition Problem 513) satisfies the differential equation $xF'=\lambda F$.

Prove that the only generalized functions on the line that are translation invariant are the constants.

Suppose that $F$ is a generalized function on the plane that is invariant under translations along the axis $Ox$.

Solve the equation $(\sin x)\cdot f(x)=0$ in the generalized functions on the line.

Suppose that the generalized function $F$ on $\mathbb {R}^{n}$ satisfies the relation $(\sum_i x_i^2-R^2)F(x)=0$ and is invariant under rotations of $\mathbb {R}^{n}$. Prove that $\langle F,\varphi \rangle =c\int_{S_R}\varphi (x)\, d\sigma (x)$, where $c$ is a constant and $d\sigma$ is the rotation invariant surface element on the sphere $S_R$ of radius $R$ in $\mathbb {R}^{n}$.

Suppose that the generalized function $F$ on the line has the following properties:

1. $F(x+1)=F(x)$;

2. $e^{2\pi ix}F(x)=F(x)$. Prove that $F(x)=c\sum_{k\in \mathbb {Z}}\delta (x-k)$.

Prove that every generalized function $F$ on the line satisfying the equation $F'(x)=a(x)F(x)+b(x)$, $a,b\in \mathcal E(\mathbb {R})$, is regular (and, hence, coincides with the ordinary solution of this equation).

Find all distributions $u \in \mathscr {D}^{\prime }\left(\mathbb {R}^{n}\right)$ that satisfy at least one of the following two conditions: