The general form of the Laguerre ODE is given by the following:

Laguerre_ode := x*diff(y(x),x,x)+(a+1-x)*diff(y(x),x)+lambda*y(x) = 0;

$\mathrm{Laguerre\_ode}≔x\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\left(a+1-x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\mathrm{\lambda }y\left(x\right)=0$

See Iyanaga and Kawada, "Encyclopedic Dictionary of Mathematics", p. 1481. The solution to this type of ODE can be expressed in terms of the WhittakerW and WhittakerM functions.

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{odeadvisor}\right)$

$\left[\mathrm{odeadvisor}\right]$

$\mathrm{odeadvisor}\left(\mathrm{Laguerre\_ode}\right)$

$\left[\mathrm{\_Laguerre}\right]$

$\mathrm{dsolve}\left(\mathrm{Laguerre\_ode}\right)$

$y\left(x\right)=\mathrm{c\_\_1}\mathrm{KummerM}\left(-\mathrm{\lambda }\,a+1\,x\right)+\mathrm{c\_\_2}\mathrm{KummerU}\left(-\mathrm{\lambda }\,a+1\,x\right)$