Solving this differential equation with an online calculator:

$$-(a z+b) y+(c z+d) y''+cy' = 0$$

I obtain something like:

$$y(z)=C_1 \exp\left(\frac{-\sqrt{a}z}{\sqrt{c}}\right) U(arg1,arg2,arg3)+C_2 \exp\left(\frac{-\sqrt{a}z}{\sqrt{c}}\right) L_{arg1}(arg2)$$

with arg are arguments of common functions.

I have two problems:

  • I don't know the functions U and L
  • if $c<0$ is it possible to find a real solution for $y(z)$ playing with the $C_1$ and $C_2$ parameters ?
  • $\begingroup$ $U$ is the confluent hypergeometric function ; $L$ is the generalized Laguerre polynomial $\endgroup$ – Claude Leibovici Nov 25 '14 at 11:09

It seems that you were using wolframalpha. There are links about the functions $U$ and $L$ - for example, Confluent Hypergeometric Function of the Second Kind and Laguerre Polynomial .

In the case where $c=0$ we have to work with Airy functions, which are, essentially, the independent solutions of the the equation $y''(z)=zy$.

As for the existence of real solutions, consider the Cauchy problem for your equation with initial data $y(z_0)=y_0$. If $cz_0+d\ne 0$, then your problem is well-posed and therefore by Cauchy-Lipschitz-Lindelof theorem has local solution, which would be real. You can build a maximum real solution afterwards. I don't know if this maximum solution will be defined on an interval, half-line or the whole line.

  • $\begingroup$ I already solved the case $c = 0$ with the Airy functions. The next step was indeed to work with the following conditions: -$c \ne 0$ -$y(z=0) = y_0$ -$y^{'}(z=z_0)=0$ I also have $c<0$ $a>0$ b and d are non-zero parameters. How to proceed ? The $C_1$ and $C_2$ constant may be foud analytically ? $\endgroup$ – user3473016 Nov 25 '14 at 11:20
  • $\begingroup$ And what if the argument 1 of the Laguerre polynomial is not an integer, which is the case in my situation ? I only use the first part of the solution ? $\endgroup$ – user3473016 Nov 25 '14 at 13:16
  • $\begingroup$ @user3473016 this, unfortunately, I don't know. There must be some generalisations of Laguerre polynomials that accept non-integer parameters. $\endgroup$ – TZakrevskiy Nov 25 '14 at 13:26

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