How to solve the following differential equation on the space of distributions? $$ u'+tu=K(0,1), $$ Here $K_{(0,1)}$ is the characteristic function of the closed interval between $0$ and $1$. I managed to find out that the problem is to find a $u$ such that the following integral equation is satisfied:
$$\int u(t) (-exp(t^2/2)) (exp(-t^2/2) p(t) )'dt=\int K_{(0,1)}(t) p(t)dt$$
where $p$ ranges over infinitely differential functions of compact support, and integration is on the line.