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I'm stuggeling with this differential equation:


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?

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What is $\Lambda$? – Davide Giraudo Feb 4 '13 at 14:55
For locally integrable function $f$ you define distribution $\Lambda_f(\phi) = \int f\phi$ . Where $\phi$ is test function$. – tom Feb 4 '13 at 14:59
Ok. But I don't the point when you write it as a sum (you could just take one constant). – Davide Giraudo Feb 4 '13 at 15:00
hmm that is true :D. – tom Feb 4 '13 at 15:14
up vote 2 down vote accepted

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.

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Is there any neat way to generalize this for any differential operator $L = \sum_{n=0}^N c_n \frac{d^n}{dx^n}$. So I'm looking for solution of $L(T) = 0$? – tom Feb 4 '13 at 16:30
I found this link… – tom Feb 4 '13 at 17:40

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