Distributions -Primitives

[Camillo]

Find all solutions $G=G(x) $ of the differential equation

$G' +a(x) G = delta $

with $ a(x) in C^(oo)(RR) $ .

[Camillo]

In order to find all solutions is useful the Theorem :

Let be $ f in ccD'(a,b) $ .Then equation $ u' = f $ in $(a,b) $ has solutions in $ccD'(a,b)$ and, if $u_0$ is one solution then all solutions are given by the distributions of the type : $ u = u_0 +C $ with $C$ an arbitrary constant ( The demonstration of such theorem is not trivial , mainly the part relevant to the existence of a solution).

Another way of solving the original equation is to multiply both sides by the integrating factor $ e^(int_0^x a(s)ds)$, which is never equal to $0$ and is $in C^(00)(RR)$.
Using also Leibniz rule, the equation becomes : $d/dx [Ge^(int_0^x a(s)ds)] = delta e^(int_0^x a(s)ds)$.
Since $ u delta = u(0)delta $, the second member is just $ delta$.
It is known that a primitive of $delta$ is $H$; consequently all primitives are given by $ H +C $.
We get than : $ Ge^(int_0^x a(s)ds) =H(x)+C $ and finally : $ G = e^(- int_0^x a(s)ds)[H(x)+C]$.

[Camillo]

The solution is correct, Eredir, but as you say there are infinite other solutions ....
Later I will indicate how to find them .

[Eredir]

Define $G = \phi_{H*f}$, where $\phi_{f}$ denotes the distribution associated to the function $f(x)$ and $H(x)$ is the Heaviside function.
Then, using the rules for the derivative of a distribution, we get $G' = \phi'_{H*f} = \phi_{(H*f)'} + \delta = \phi_{H*f'} + \delta$.
Plugging this in the differential equation gives $\phi_{H*f'} + a(x)\phi_{H*f} = 0$.

We can apply this distribution to some test function $z(x)$.
We have $\phi_{H*f'}(z) + a(x)\phi_{H*f}(z) = \int_{-\infty}^{+\infty}H(x)*f'(x)*z(x)dx + \int_{-\infty}^{+\infty}a(x)*H(x)*f(x)*z(x)dx$
$ = \int_0^{+\infty}[f'(x) + a(x)*f(x)]z(x)dx = 0$.

Since this is valid for any test function $z(x)$ we have $f'(x) + a(x)*f(x) = 0$.
The solution is simply $f(x) = e^{-A(x)}$, where $A(x)$ is a primitive of $a(x)$.

Then a solution of the initial differential equation is the distribution $G = \phi_{H*e^{-A}}$, that is the distribution associated to the function $H(x)*e^{-A(x)}$.

I'm not really that knowledgeable in this subject, I hope this makes some sense.
If this is correct it still remains to show if other solutions exist or not.