I Is there a name for this sort of differential equation?

[Gear300]

Is there a name to this sort of differential equation?
$$
f(z) + 2zf'(z) + f''(z) = 0 ~.
$$
I ran into it somewhere and it does not look to be Hermite. I think it has the general solution
$$
f(z) = e^{-z^2} \big( c_1 + c_2 \Phi(\sqrt{3}z) \big)
\quad \textnormal{($\Phi(x)$ is probit function.)}
$$
You might have to correct me on the solution, but is there a name to it?

 

[topsquark]

Is there a name to this sort of differential equation?
$$
f(z) + 2zf'(z) + f''(z) = 0 ~.
$$
I ran into it somewhere and it does not look to be Hermite. I think it has the general solution
$$
f(z) = e^{-z^2} \big( c_1 + c_2 \Phi(\sqrt{3}z) \big)
\quad \textnormal{($\Phi(x)$ is probit function.)}
$$
You might have to correct me on the solution, but is there a name to it?

This can be transformed into
$\dfrac{d}{dz} \left ( e^{z^2} \dfrac{df}{dz} \right ) + e^{z^2} f(z) = 0$

This is a Sturm-Louville differential equation.

-Dan

 

[Gear300]

Ah. Thanks.