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

AI Thread Summary
The differential equation presented, f(z) + 2zf'(z) + f''(z) = 0, does not appear to be classified as a Hermite equation. The suggested general solution involves the probit function, expressed as f(z) = e^{-z^2} (c_1 + c_2 Φ(√3z)). It can be transformed into a Sturm-Liouville form, which is a recognized category of differential equations. This transformation confirms its classification within that framework. The discussion concludes with acknowledgment of the Sturm-Liouville connection.
Gear300
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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?
 
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Gear300 said:
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
 
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Ah. Thanks.
 
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