Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \forall x\in\mathbb{R}.$$ Given $0<\sigma\le 1$. I wish to know whether there exists an explicit solution $\alpha$ satisfying the differential equation $$\alpha''(u) [\phi(u) + \frac 1\sigma \phi(u/\sigma)] - \alpha'(u) [u\phi(u) + \frac 1{\sigma^3} u\phi(u/\sigma)] - \alpha(u) [\phi(u) + \frac 1{\sigma^3} \phi(u/\sigma)] = \frac 1{\sigma^5} u\phi(u/\sigma).$$
For $\sigma=1$, I got that the explicit solution for the above differential equation is $\alpha(u)= -\frac 14 u + (a + b \Phi(u))/\phi(u)$ where $\Phi(u)=\int_{-\infty}^u\phi(x)dx.$ However, when $\sigma<1$ it really makes difficult in finding the solution. Any suggestion?
After correcting the mistake in my first answer, the method of solving fails. So my answer must be withdrawn.