Let $a\neq 0$, if one assume $u$ is a solution of the following differential equation on $(0,\infty)$, $$au'+b u=0.$$ Then define $$v(x)= \begin{cases}u(x) & x >0 \\ u(-x) & x <0 \end{cases}$$ and $$w(x)= \begin{cases}u(x) & x >0 \\ -u(-x) & x <0 \end{cases}.$$ We can get the distribution equation $aV'(v)+bV(v)=0$ and $aW'(w)+bW(w)=0.$
My question is : How one can obtain a classical solution of the initial value problem from the distribution theory? e.g.$$au'+b u=0,x>0; u(0)=\alpha.$$