What are the most general distributional solutions $u \in \mathcal{D}'(\mathbb{R})$ to
- $-\frac{d^n u}{dx^n} + c_{n-1}\frac{d^{n-1}u}{dx^{n-1}} + ... + c_0 u = 0$;
- $-x\frac{d^n u}{dx^n} + c_{n-1}x\frac{d^{n-1}u}{dx^{n-1}} + ... + c_0x u = 0$;
where the $c_i$'s are constants.
For the first one, using the definition of the distributional derivative, I came to the conclusion that it suffices to solve the $n$th-order characteristic polynomial and to fix $u =$ continuous function that is solution to differential operator $-\frac{d^n }{dx^n} + c_{n-1}\frac{d^{n-1}}{dx^{n-1}} + ... + c_0 = 0$. Is it OK ?
As for the second one I'm stuck... What is the general strategy to solve a problem of this kind ?
Solutions will be just smooth solutions of $Lu=-\frac{d^n u}{dx^n} + c_{n-1}\frac{d^{n-1}u}{dx^{n-1}} + ... + c_0 u = 0$. Yes, they can be obtained by solving the characteristic equation etc.
Solutions satisfy $Lu=f$, where $xf=0$ in $\mathcal{D}'(\mathbb{R})$. It follows that $\operatorname{supp} f=\{0\}$. It is known that only distributions with a point support are (finite) linear combinations of the delta-function and its derivatives: $f(x)=\sum_{k=0}^m a_k \delta^{(k)}(x)$. The equation $xf=0$ is satisfied only by $f(x)=a\delta(x)\!$. So the equation becomes $Lu(x)=a\delta(x)$ and a general solution is $u_0+aG$, where $u_0$ is a general solution of the homogeneous equation $Lu=0$ and $G$ is a fundamental solution of the operator $L$.