I'm stuggeling with this differential equation:
$T'+T=0$
Where $T$ is distribution.
I found solutions in form:
$\sum_{n\in A} \frac{d^n}{dx^n}\Lambda_{c_n e^{-x}}$. This can be simplified to $\sum_{n\in A} \Lambda_{b_n e^{-x}}$.
Where $A \subset \mathbb{N}_0$ finite, $c_n,b_n$ arbitrary. But I don't know if I found all solutions.
Can anybody help me, please?
Assume that $T$ is solution, and let $S:=e^tT$, that is the distribution defined by $\langle S,x\mapsto \phi(x)\rangle:=\langle T,x\mapsto e^x\phi(x)\rangle$. We have $$S'(\phi)=-S(\phi')=-T(e^t\phi')=-T((e^t\phi)'-e^t\phi)=-T(e^t\phi)+T(e^t\phi)=0.$$ Conclude by this thread.