Laguerre ODEs - Maple Help

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Laguerre ODEs

Description

 • 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;
 ${\mathrm{Laguerre_ode}}{:=}{x}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right)\right){+}\left({a}{+}{1}{-}{x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){+}{\mathrm{λ}}{}{y}{}\left({x}\right){=}{0}$ (1)
 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.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$
 $\left[{\mathrm{odeadvisor}}\right]$ (2)
 > $\mathrm{odeadvisor}\left(\mathrm{Laguerre_ode}\right)$
 $\left[{\mathrm{_Laguerre}}\right]$ (3)
 > $\mathrm{dsolve}\left(\mathrm{Laguerre_ode}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{KummerM}}{}\left({-}{\mathrm{λ}}{,}{a}{+}{1}{,}{x}\right){+}{\mathrm{_C2}}{}{\mathrm{KummerU}}{}\left({-}{\mathrm{λ}}{,}{a}{+}{1}{,}{x}\right)$ (4)
 See Also DEtools, odeadvisor, dsolve, and ?odeadvisor, where is one of: quadrature, missing, reducible, linear_ODEs, exact_linear, exact_nonlinear, sym_Fx, linear_sym, Bessel, Painleve, Halm, Gegenbauer, Duffing, ellipsoidal, elliptic, erf, Emden, Jacobi, Hermite, Lagerstrom, Laguerre, Liouville, Lienard, Van_der_Pol, Titchmarsh; for other differential orders see odeadvisor,types.